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# Polytope of Type {4,4,4}

Atlas Canonical Name : {4,4,4}*512a
if this polytope has a name.
Group : SmallGroup(512,152421)
Rank : 4
Schlafli Type : {4,4,4}
Number of vertices, edges, etc : 8, 32, 32, 8
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Locally Toroidal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,4}*256a, {4,4,4}*256b
4-fold quotients : {4,4,4}*128, {2,4,4}*128, {4,4,2}*128
8-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
16-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
32-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 43)( 10, 44)( 11, 41)( 12, 42)( 13, 47)( 14, 48)( 15, 45)( 16, 46)
( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)( 24, 56)
( 25, 59)( 26, 60)( 27, 57)( 28, 58)( 29, 63)( 30, 64)( 31, 61)( 32, 62)
( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)( 72,104)
( 73,107)( 74,108)( 75,105)( 76,106)( 77,111)( 78,112)( 79,109)( 80,110)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)( 88,120)
( 89,123)( 90,124)( 91,121)( 92,122)( 93,127)( 94,128)( 95,125)( 96,126)
(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)
(137,171)(138,172)(139,169)(140,170)(141,175)(142,176)(143,173)(144,174)
(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)(152,184)
(153,187)(154,188)(155,185)(156,186)(157,191)(158,192)(159,189)(160,190)
(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)(200,232)
(201,235)(202,236)(203,233)(204,234)(205,239)(206,240)(207,237)(208,238)
(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)(216,248)
(217,251)(218,252)(219,249)(220,250)(221,255)(222,256)(223,253)(224,254)
(257,289)(258,290)(259,291)(260,292)(261,293)(262,294)(263,295)(264,296)
(265,299)(266,300)(267,297)(268,298)(269,303)(270,304)(271,301)(272,302)
(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)(280,312)
(281,315)(282,316)(283,313)(284,314)(285,319)(286,320)(287,317)(288,318)
(321,353)(322,354)(323,355)(324,356)(325,357)(326,358)(327,359)(328,360)
(329,363)(330,364)(331,361)(332,362)(333,367)(334,368)(335,365)(336,366)
(337,369)(338,370)(339,371)(340,372)(341,373)(342,374)(343,375)(344,376)
(345,379)(346,380)(347,377)(348,378)(349,383)(350,384)(351,381)(352,382)
(385,417)(386,418)(387,419)(388,420)(389,421)(390,422)(391,423)(392,424)
(393,427)(394,428)(395,425)(396,426)(397,431)(398,432)(399,429)(400,430)
(401,433)(402,434)(403,435)(404,436)(405,437)(406,438)(407,439)(408,440)
(409,443)(410,444)(411,441)(412,442)(413,447)(414,448)(415,445)(416,446)
(449,481)(450,482)(451,483)(452,484)(453,485)(454,486)(455,487)(456,488)
(457,491)(458,492)(459,489)(460,490)(461,495)(462,496)(463,493)(464,494)
(465,497)(466,498)(467,499)(468,500)(469,501)(470,502)(471,503)(472,504)
(473,507)(474,508)(475,505)(476,506)(477,511)(478,512)(479,509)(480,510);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,264)(  6,263)(  7,262)(  8,261)
(  9,265)( 10,266)( 11,267)( 12,268)( 13,272)( 14,271)( 15,270)( 16,269)
( 17,273)( 18,274)( 19,275)( 20,276)( 21,280)( 22,279)( 23,278)( 24,277)
( 25,281)( 26,282)( 27,283)( 28,284)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,305)( 34,306)( 35,307)( 36,308)( 37,312)( 38,311)( 39,310)( 40,309)
( 41,313)( 42,314)( 43,315)( 44,316)( 45,320)( 46,319)( 47,318)( 48,317)
( 49,289)( 50,290)( 51,291)( 52,292)( 53,296)( 54,295)( 55,294)( 56,293)
( 57,297)( 58,298)( 59,299)( 60,300)( 61,304)( 62,303)( 63,302)( 64,301)
( 65,329)( 66,330)( 67,331)( 68,332)( 69,336)( 70,335)( 71,334)( 72,333)
( 73,321)( 74,322)( 75,323)( 76,324)( 77,328)( 78,327)( 79,326)( 80,325)
( 81,345)( 82,346)( 83,347)( 84,348)( 85,352)( 86,351)( 87,350)( 88,349)
( 89,337)( 90,338)( 91,339)( 92,340)( 93,344)( 94,343)( 95,342)( 96,341)
( 97,379)( 98,380)( 99,377)(100,378)(101,382)(102,381)(103,384)(104,383)
(105,371)(106,372)(107,369)(108,370)(109,374)(110,373)(111,376)(112,375)
(113,363)(114,364)(115,361)(116,362)(117,366)(118,365)(119,368)(120,367)
(121,355)(122,356)(123,353)(124,354)(125,358)(126,357)(127,360)(128,359)
(129,401)(130,402)(131,403)(132,404)(133,408)(134,407)(135,406)(136,405)
(137,409)(138,410)(139,411)(140,412)(141,416)(142,415)(143,414)(144,413)
(145,385)(146,386)(147,387)(148,388)(149,392)(150,391)(151,390)(152,389)
(153,393)(154,394)(155,395)(156,396)(157,400)(158,399)(159,398)(160,397)
(161,417)(162,418)(163,419)(164,420)(165,424)(166,423)(167,422)(168,421)
(169,425)(170,426)(171,427)(172,428)(173,432)(174,431)(175,430)(176,429)
(177,433)(178,434)(179,435)(180,436)(181,440)(182,439)(183,438)(184,437)
(185,441)(186,442)(187,443)(188,444)(189,448)(190,447)(191,446)(192,445)
(193,475)(194,476)(195,473)(196,474)(197,478)(198,477)(199,480)(200,479)
(201,467)(202,468)(203,465)(204,466)(205,470)(206,469)(207,472)(208,471)
(209,459)(210,460)(211,457)(212,458)(213,462)(214,461)(215,464)(216,463)
(217,451)(218,452)(219,449)(220,450)(221,454)(222,453)(223,456)(224,455)
(225,489)(226,490)(227,491)(228,492)(229,496)(230,495)(231,494)(232,493)
(233,481)(234,482)(235,483)(236,484)(237,488)(238,487)(239,486)(240,485)
(241,505)(242,506)(243,507)(244,508)(245,512)(246,511)(247,510)(248,509)
(249,497)(250,498)(251,499)(252,500)(253,504)(254,503)(255,502)(256,501);;
s2 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 87)( 22, 88)( 23, 85)( 24, 86)
( 25, 91)( 26, 92)( 27, 89)( 28, 90)( 29, 95)( 30, 96)( 31, 93)( 32, 94)
( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,101)( 38,102)( 39,103)( 40,104)
( 41,105)( 42,106)( 43,107)( 44,108)( 45,109)( 46,110)( 47,111)( 48,112)
( 49,115)( 50,116)( 51,113)( 52,114)( 53,119)( 54,120)( 55,117)( 56,118)
( 57,123)( 58,124)( 59,121)( 60,122)( 61,127)( 62,128)( 63,125)( 64,126)
(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)(136,200)
(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)
(145,211)(146,212)(147,209)(148,210)(149,215)(150,216)(151,213)(152,214)
(153,219)(154,220)(155,217)(156,218)(157,223)(158,224)(159,221)(160,222)
(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)(168,232)
(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)
(177,243)(178,244)(179,241)(180,242)(181,247)(182,248)(183,245)(184,246)
(185,251)(186,252)(187,249)(188,250)(189,255)(190,256)(191,253)(192,254)
(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)(264,328)
(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)(272,336)
(273,339)(274,340)(275,337)(276,338)(277,343)(278,344)(279,341)(280,342)
(281,347)(282,348)(283,345)(284,346)(285,351)(286,352)(287,349)(288,350)
(289,353)(290,354)(291,355)(292,356)(293,357)(294,358)(295,359)(296,360)
(297,361)(298,362)(299,363)(300,364)(301,365)(302,366)(303,367)(304,368)
(305,371)(306,372)(307,369)(308,370)(309,375)(310,376)(311,373)(312,374)
(313,379)(314,380)(315,377)(316,378)(317,383)(318,384)(319,381)(320,382)
(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)(392,456)
(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)(400,464)
(401,467)(402,468)(403,465)(404,466)(405,471)(406,472)(407,469)(408,470)
(409,475)(410,476)(411,473)(412,474)(413,479)(414,480)(415,477)(416,478)
(417,481)(418,482)(419,483)(420,484)(421,485)(422,486)(423,487)(424,488)
(425,489)(426,490)(427,491)(428,492)(429,493)(430,494)(431,495)(432,496)
(433,499)(434,500)(435,497)(436,498)(437,503)(438,504)(439,501)(440,502)
(441,507)(442,508)(443,505)(444,506)(445,511)(446,512)(447,509)(448,510);;
s3 := (  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)(  8,168)
(  9,172)( 10,171)( 11,170)( 12,169)( 13,176)( 14,175)( 15,174)( 16,173)
( 17,177)( 18,178)( 19,179)( 20,180)( 21,181)( 22,182)( 23,183)( 24,184)
( 25,188)( 26,187)( 27,186)( 28,185)( 29,192)( 30,191)( 31,190)( 32,189)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,140)( 42,139)( 43,138)( 44,137)( 45,144)( 46,143)( 47,142)( 48,141)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,156)( 58,155)( 59,154)( 60,153)( 61,160)( 62,159)( 63,158)( 64,157)
( 65,229)( 66,230)( 67,231)( 68,232)( 69,225)( 70,226)( 71,227)( 72,228)
( 73,240)( 74,239)( 75,238)( 76,237)( 77,236)( 78,235)( 79,234)( 80,233)
( 81,245)( 82,246)( 83,247)( 84,248)( 85,241)( 86,242)( 87,243)( 88,244)
( 89,256)( 90,255)( 91,254)( 92,253)( 93,252)( 94,251)( 95,250)( 96,249)
( 97,197)( 98,198)( 99,199)(100,200)(101,193)(102,194)(103,195)(104,196)
(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201)
(113,213)(114,214)(115,215)(116,216)(117,209)(118,210)(119,211)(120,212)
(121,224)(122,223)(123,222)(124,221)(125,220)(126,219)(127,218)(128,217)
(257,417)(258,418)(259,419)(260,420)(261,421)(262,422)(263,423)(264,424)
(265,428)(266,427)(267,426)(268,425)(269,432)(270,431)(271,430)(272,429)
(273,433)(274,434)(275,435)(276,436)(277,437)(278,438)(279,439)(280,440)
(281,444)(282,443)(283,442)(284,441)(285,448)(286,447)(287,446)(288,445)
(289,385)(290,386)(291,387)(292,388)(293,389)(294,390)(295,391)(296,392)
(297,396)(298,395)(299,394)(300,393)(301,400)(302,399)(303,398)(304,397)
(305,401)(306,402)(307,403)(308,404)(309,405)(310,406)(311,407)(312,408)
(313,412)(314,411)(315,410)(316,409)(317,416)(318,415)(319,414)(320,413)
(321,485)(322,486)(323,487)(324,488)(325,481)(326,482)(327,483)(328,484)
(329,496)(330,495)(331,494)(332,493)(333,492)(334,491)(335,490)(336,489)
(337,501)(338,502)(339,503)(340,504)(341,497)(342,498)(343,499)(344,500)
(345,512)(346,511)(347,510)(348,509)(349,508)(350,507)(351,506)(352,505)
(353,453)(354,454)(355,455)(356,456)(357,449)(358,450)(359,451)(360,452)
(361,464)(362,463)(363,462)(364,461)(365,460)(366,459)(367,458)(368,457)
(369,469)(370,470)(371,471)(372,472)(373,465)(374,466)(375,467)(376,468)
(377,480)(378,479)(379,478)(380,477)(381,476)(382,475)(383,474)(384,473);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(512)!(  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)
(  8, 40)(  9, 43)( 10, 44)( 11, 41)( 12, 42)( 13, 47)( 14, 48)( 15, 45)
( 16, 46)( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)
( 24, 56)( 25, 59)( 26, 60)( 27, 57)( 28, 58)( 29, 63)( 30, 64)( 31, 61)
( 32, 62)( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)
( 72,104)( 73,107)( 74,108)( 75,105)( 76,106)( 77,111)( 78,112)( 79,109)
( 80,110)( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)
( 88,120)( 89,123)( 90,124)( 91,121)( 92,122)( 93,127)( 94,128)( 95,125)
( 96,126)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)
(136,168)(137,171)(138,172)(139,169)(140,170)(141,175)(142,176)(143,173)
(144,174)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)
(152,184)(153,187)(154,188)(155,185)(156,186)(157,191)(158,192)(159,189)
(160,190)(193,225)(194,226)(195,227)(196,228)(197,229)(198,230)(199,231)
(200,232)(201,235)(202,236)(203,233)(204,234)(205,239)(206,240)(207,237)
(208,238)(209,241)(210,242)(211,243)(212,244)(213,245)(214,246)(215,247)
(216,248)(217,251)(218,252)(219,249)(220,250)(221,255)(222,256)(223,253)
(224,254)(257,289)(258,290)(259,291)(260,292)(261,293)(262,294)(263,295)
(264,296)(265,299)(266,300)(267,297)(268,298)(269,303)(270,304)(271,301)
(272,302)(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)
(280,312)(281,315)(282,316)(283,313)(284,314)(285,319)(286,320)(287,317)
(288,318)(321,353)(322,354)(323,355)(324,356)(325,357)(326,358)(327,359)
(328,360)(329,363)(330,364)(331,361)(332,362)(333,367)(334,368)(335,365)
(336,366)(337,369)(338,370)(339,371)(340,372)(341,373)(342,374)(343,375)
(344,376)(345,379)(346,380)(347,377)(348,378)(349,383)(350,384)(351,381)
(352,382)(385,417)(386,418)(387,419)(388,420)(389,421)(390,422)(391,423)
(392,424)(393,427)(394,428)(395,425)(396,426)(397,431)(398,432)(399,429)
(400,430)(401,433)(402,434)(403,435)(404,436)(405,437)(406,438)(407,439)
(408,440)(409,443)(410,444)(411,441)(412,442)(413,447)(414,448)(415,445)
(416,446)(449,481)(450,482)(451,483)(452,484)(453,485)(454,486)(455,487)
(456,488)(457,491)(458,492)(459,489)(460,490)(461,495)(462,496)(463,493)
(464,494)(465,497)(466,498)(467,499)(468,500)(469,501)(470,502)(471,503)
(472,504)(473,507)(474,508)(475,505)(476,506)(477,511)(478,512)(479,509)
(480,510);
s1 := Sym(512)!(  1,257)(  2,258)(  3,259)(  4,260)(  5,264)(  6,263)(  7,262)
(  8,261)(  9,265)( 10,266)( 11,267)( 12,268)( 13,272)( 14,271)( 15,270)
( 16,269)( 17,273)( 18,274)( 19,275)( 20,276)( 21,280)( 22,279)( 23,278)
( 24,277)( 25,281)( 26,282)( 27,283)( 28,284)( 29,288)( 30,287)( 31,286)
( 32,285)( 33,305)( 34,306)( 35,307)( 36,308)( 37,312)( 38,311)( 39,310)
( 40,309)( 41,313)( 42,314)( 43,315)( 44,316)( 45,320)( 46,319)( 47,318)
( 48,317)( 49,289)( 50,290)( 51,291)( 52,292)( 53,296)( 54,295)( 55,294)
( 56,293)( 57,297)( 58,298)( 59,299)( 60,300)( 61,304)( 62,303)( 63,302)
( 64,301)( 65,329)( 66,330)( 67,331)( 68,332)( 69,336)( 70,335)( 71,334)
( 72,333)( 73,321)( 74,322)( 75,323)( 76,324)( 77,328)( 78,327)( 79,326)
( 80,325)( 81,345)( 82,346)( 83,347)( 84,348)( 85,352)( 86,351)( 87,350)
( 88,349)( 89,337)( 90,338)( 91,339)( 92,340)( 93,344)( 94,343)( 95,342)
( 96,341)( 97,379)( 98,380)( 99,377)(100,378)(101,382)(102,381)(103,384)
(104,383)(105,371)(106,372)(107,369)(108,370)(109,374)(110,373)(111,376)
(112,375)(113,363)(114,364)(115,361)(116,362)(117,366)(118,365)(119,368)
(120,367)(121,355)(122,356)(123,353)(124,354)(125,358)(126,357)(127,360)
(128,359)(129,401)(130,402)(131,403)(132,404)(133,408)(134,407)(135,406)
(136,405)(137,409)(138,410)(139,411)(140,412)(141,416)(142,415)(143,414)
(144,413)(145,385)(146,386)(147,387)(148,388)(149,392)(150,391)(151,390)
(152,389)(153,393)(154,394)(155,395)(156,396)(157,400)(158,399)(159,398)
(160,397)(161,417)(162,418)(163,419)(164,420)(165,424)(166,423)(167,422)
(168,421)(169,425)(170,426)(171,427)(172,428)(173,432)(174,431)(175,430)
(176,429)(177,433)(178,434)(179,435)(180,436)(181,440)(182,439)(183,438)
(184,437)(185,441)(186,442)(187,443)(188,444)(189,448)(190,447)(191,446)
(192,445)(193,475)(194,476)(195,473)(196,474)(197,478)(198,477)(199,480)
(200,479)(201,467)(202,468)(203,465)(204,466)(205,470)(206,469)(207,472)
(208,471)(209,459)(210,460)(211,457)(212,458)(213,462)(214,461)(215,464)
(216,463)(217,451)(218,452)(219,449)(220,450)(221,454)(222,453)(223,456)
(224,455)(225,489)(226,490)(227,491)(228,492)(229,496)(230,495)(231,494)
(232,493)(233,481)(234,482)(235,483)(236,484)(237,488)(238,487)(239,486)
(240,485)(241,505)(242,506)(243,507)(244,508)(245,512)(246,511)(247,510)
(248,509)(249,497)(250,498)(251,499)(252,500)(253,504)(254,503)(255,502)
(256,501);
s2 := Sym(512)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)
(  8, 72)(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 87)( 22, 88)( 23, 85)
( 24, 86)( 25, 91)( 26, 92)( 27, 89)( 28, 90)( 29, 95)( 30, 96)( 31, 93)
( 32, 94)( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,101)( 38,102)( 39,103)
( 40,104)( 41,105)( 42,106)( 43,107)( 44,108)( 45,109)( 46,110)( 47,111)
( 48,112)( 49,115)( 50,116)( 51,113)( 52,114)( 53,119)( 54,120)( 55,117)
( 56,118)( 57,123)( 58,124)( 59,121)( 60,122)( 61,127)( 62,128)( 63,125)
( 64,126)(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)
(136,200)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)
(144,208)(145,211)(146,212)(147,209)(148,210)(149,215)(150,216)(151,213)
(152,214)(153,219)(154,220)(155,217)(156,218)(157,223)(158,224)(159,221)
(160,222)(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)
(168,232)(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)
(176,240)(177,243)(178,244)(179,241)(180,242)(181,247)(182,248)(183,245)
(184,246)(185,251)(186,252)(187,249)(188,250)(189,255)(190,256)(191,253)
(192,254)(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)
(264,328)(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)
(272,336)(273,339)(274,340)(275,337)(276,338)(277,343)(278,344)(279,341)
(280,342)(281,347)(282,348)(283,345)(284,346)(285,351)(286,352)(287,349)
(288,350)(289,353)(290,354)(291,355)(292,356)(293,357)(294,358)(295,359)
(296,360)(297,361)(298,362)(299,363)(300,364)(301,365)(302,366)(303,367)
(304,368)(305,371)(306,372)(307,369)(308,370)(309,375)(310,376)(311,373)
(312,374)(313,379)(314,380)(315,377)(316,378)(317,383)(318,384)(319,381)
(320,382)(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)
(392,456)(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)
(400,464)(401,467)(402,468)(403,465)(404,466)(405,471)(406,472)(407,469)
(408,470)(409,475)(410,476)(411,473)(412,474)(413,479)(414,480)(415,477)
(416,478)(417,481)(418,482)(419,483)(420,484)(421,485)(422,486)(423,487)
(424,488)(425,489)(426,490)(427,491)(428,492)(429,493)(430,494)(431,495)
(432,496)(433,499)(434,500)(435,497)(436,498)(437,503)(438,504)(439,501)
(440,502)(441,507)(442,508)(443,505)(444,506)(445,511)(446,512)(447,509)
(448,510);
s3 := Sym(512)!(  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)
(  8,168)(  9,172)( 10,171)( 11,170)( 12,169)( 13,176)( 14,175)( 15,174)
( 16,173)( 17,177)( 18,178)( 19,179)( 20,180)( 21,181)( 22,182)( 23,183)
( 24,184)( 25,188)( 26,187)( 27,186)( 28,185)( 29,192)( 30,191)( 31,190)
( 32,189)( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,140)( 42,139)( 43,138)( 44,137)( 45,144)( 46,143)( 47,142)
( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)
( 56,152)( 57,156)( 58,155)( 59,154)( 60,153)( 61,160)( 62,159)( 63,158)
( 64,157)( 65,229)( 66,230)( 67,231)( 68,232)( 69,225)( 70,226)( 71,227)
( 72,228)( 73,240)( 74,239)( 75,238)( 76,237)( 77,236)( 78,235)( 79,234)
( 80,233)( 81,245)( 82,246)( 83,247)( 84,248)( 85,241)( 86,242)( 87,243)
( 88,244)( 89,256)( 90,255)( 91,254)( 92,253)( 93,252)( 94,251)( 95,250)
( 96,249)( 97,197)( 98,198)( 99,199)(100,200)(101,193)(102,194)(103,195)
(104,196)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)
(112,201)(113,213)(114,214)(115,215)(116,216)(117,209)(118,210)(119,211)
(120,212)(121,224)(122,223)(123,222)(124,221)(125,220)(126,219)(127,218)
(128,217)(257,417)(258,418)(259,419)(260,420)(261,421)(262,422)(263,423)
(264,424)(265,428)(266,427)(267,426)(268,425)(269,432)(270,431)(271,430)
(272,429)(273,433)(274,434)(275,435)(276,436)(277,437)(278,438)(279,439)
(280,440)(281,444)(282,443)(283,442)(284,441)(285,448)(286,447)(287,446)
(288,445)(289,385)(290,386)(291,387)(292,388)(293,389)(294,390)(295,391)
(296,392)(297,396)(298,395)(299,394)(300,393)(301,400)(302,399)(303,398)
(304,397)(305,401)(306,402)(307,403)(308,404)(309,405)(310,406)(311,407)
(312,408)(313,412)(314,411)(315,410)(316,409)(317,416)(318,415)(319,414)
(320,413)(321,485)(322,486)(323,487)(324,488)(325,481)(326,482)(327,483)
(328,484)(329,496)(330,495)(331,494)(332,493)(333,492)(334,491)(335,490)
(336,489)(337,501)(338,502)(339,503)(340,504)(341,497)(342,498)(343,499)
(344,500)(345,512)(346,511)(347,510)(348,509)(349,508)(350,507)(351,506)
(352,505)(353,453)(354,454)(355,455)(356,456)(357,449)(358,450)(359,451)
(360,452)(361,464)(362,463)(363,462)(364,461)(365,460)(366,459)(367,458)
(368,457)(369,469)(370,470)(371,471)(372,472)(373,465)(374,466)(375,467)
(376,468)(377,480)(378,479)(379,478)(380,477)(381,476)(382,475)(383,474)
(384,473);
poly := sub<Sym(512)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1 >;

```
References : None.
to this polytope