Polytope of Type {8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4}*512a
if this polytope has a name.
Group : SmallGroup(512,58326)
Rank : 3
Schlafli Type : {8,4}
Number of vertices, edges, etc : 64, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,4}*256a
4-fold quotients : {8,4}*128a, {4,4}*128, {8,4}*128b
8-fold quotients : {8,4}*64a, {8,4}*64b, {4,4}*64
16-fold quotients : {4,4}*32, {8,2}*32
32-fold quotients : {2,4}*16, {4,2}*16
64-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,129)( 2,130)( 3,132)( 4,131)( 5,134)( 6,133)( 7,135)( 8,136)
( 9,137)( 10,138)( 11,140)( 12,139)( 13,142)( 14,141)( 15,143)( 16,144)
( 17,151)( 18,152)( 19,150)( 20,149)( 21,148)( 22,147)( 23,145)( 24,146)
( 25,159)( 26,160)( 27,158)( 28,157)( 29,156)( 30,155)( 31,153)( 32,154)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,174)( 38,173)( 39,175)( 40,176)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,166)( 46,165)( 47,167)( 48,168)
( 49,191)( 50,192)( 51,190)( 52,189)( 53,188)( 54,187)( 55,185)( 56,186)
( 57,183)( 58,184)( 59,182)( 60,181)( 61,180)( 62,179)( 63,177)( 64,178)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,198)( 70,197)( 71,199)( 72,200)
( 73,201)( 74,202)( 75,204)( 76,203)( 77,206)( 78,205)( 79,207)( 80,208)
( 81,215)( 82,216)( 83,214)( 84,213)( 85,212)( 86,211)( 87,209)( 88,210)
( 89,223)( 90,224)( 91,222)( 92,221)( 93,220)( 94,219)( 95,217)( 96,218)
( 97,233)( 98,234)( 99,236)(100,235)(101,238)(102,237)(103,239)(104,240)
(105,225)(106,226)(107,228)(108,227)(109,230)(110,229)(111,231)(112,232)
(113,255)(114,256)(115,254)(116,253)(117,252)(118,251)(119,249)(120,250)
(121,247)(122,248)(123,246)(124,245)(125,244)(126,243)(127,241)(128,242)
(257,385)(258,386)(259,388)(260,387)(261,390)(262,389)(263,391)(264,392)
(265,393)(266,394)(267,396)(268,395)(269,398)(270,397)(271,399)(272,400)
(273,407)(274,408)(275,406)(276,405)(277,404)(278,403)(279,401)(280,402)
(281,415)(282,416)(283,414)(284,413)(285,412)(286,411)(287,409)(288,410)
(289,425)(290,426)(291,428)(292,427)(293,430)(294,429)(295,431)(296,432)
(297,417)(298,418)(299,420)(300,419)(301,422)(302,421)(303,423)(304,424)
(305,447)(306,448)(307,446)(308,445)(309,444)(310,443)(311,441)(312,442)
(313,439)(314,440)(315,438)(316,437)(317,436)(318,435)(319,433)(320,434)
(321,449)(322,450)(323,452)(324,451)(325,454)(326,453)(327,455)(328,456)
(329,457)(330,458)(331,460)(332,459)(333,462)(334,461)(335,463)(336,464)
(337,471)(338,472)(339,470)(340,469)(341,468)(342,467)(343,465)(344,466)
(345,479)(346,480)(347,478)(348,477)(349,476)(350,475)(351,473)(352,474)
(353,489)(354,490)(355,492)(356,491)(357,494)(358,493)(359,495)(360,496)
(361,481)(362,482)(363,484)(364,483)(365,486)(366,485)(367,487)(368,488)
(369,511)(370,512)(371,510)(372,509)(373,508)(374,507)(375,505)(376,506)
(377,503)(378,504)(379,502)(380,501)(381,500)(382,499)(383,497)(384,498);;
s1 := ( 1,257)( 2,258)( 3,259)( 4,260)( 5,263)( 6,264)( 7,261)( 8,262)
( 9,265)( 10,266)( 11,267)( 12,268)( 13,271)( 14,272)( 15,269)( 16,270)
( 17,273)( 18,274)( 19,275)( 20,276)( 21,279)( 22,280)( 23,277)( 24,278)
( 25,281)( 26,282)( 27,283)( 28,284)( 29,287)( 30,288)( 31,285)( 32,286)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,303)( 38,304)( 39,301)( 40,302)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,295)( 46,296)( 47,293)( 48,294)
( 49,313)( 50,314)( 51,315)( 52,316)( 53,319)( 54,320)( 55,317)( 56,318)
( 57,305)( 58,306)( 59,307)( 60,308)( 61,311)( 62,312)( 63,309)( 64,310)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)( 72,342)
( 73,345)( 74,346)( 75,347)( 76,348)( 77,351)( 78,352)( 79,349)( 80,350)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)( 88,326)
( 89,329)( 90,330)( 91,331)( 92,332)( 93,335)( 94,336)( 95,333)( 96,334)
( 97,380)( 98,379)( 99,378)(100,377)(101,382)(102,381)(103,384)(104,383)
(105,372)(106,371)(107,370)(108,369)(109,374)(110,373)(111,376)(112,375)
(113,364)(114,363)(115,362)(116,361)(117,366)(118,365)(119,368)(120,367)
(121,356)(122,355)(123,354)(124,353)(125,358)(126,357)(127,360)(128,359)
(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)(136,422)
(137,425)(138,426)(139,427)(140,428)(141,431)(142,432)(143,429)(144,430)
(145,433)(146,434)(147,435)(148,436)(149,439)(150,440)(151,437)(152,438)
(153,441)(154,442)(155,443)(156,444)(157,447)(158,448)(159,445)(160,446)
(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)(168,390)
(169,393)(170,394)(171,395)(172,396)(173,399)(174,400)(175,397)(176,398)
(177,401)(178,402)(179,403)(180,404)(181,407)(182,408)(183,405)(184,406)
(185,409)(186,410)(187,411)(188,412)(189,415)(190,416)(191,413)(192,414)
(193,501)(194,502)(195,503)(196,504)(197,499)(198,500)(199,497)(200,498)
(201,509)(202,510)(203,511)(204,512)(205,507)(206,508)(207,505)(208,506)
(209,485)(210,486)(211,487)(212,488)(213,483)(214,484)(215,481)(216,482)
(217,493)(218,494)(219,495)(220,496)(221,491)(222,492)(223,489)(224,490)
(225,471)(226,472)(227,469)(228,470)(229,465)(230,466)(231,467)(232,468)
(233,479)(234,480)(235,477)(236,478)(237,473)(238,474)(239,475)(240,476)
(241,455)(242,456)(243,453)(244,454)(245,449)(246,450)(247,451)(248,452)
(249,463)(250,464)(251,461)(252,462)(253,457)(254,458)(255,459)(256,460);;
s2 := ( 1, 65)( 2, 66)( 3, 67)( 4, 68)( 5, 69)( 6, 70)( 7, 71)( 8, 72)
( 9, 74)( 10, 73)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 80)( 16, 79)
( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)( 24, 88)
( 25, 90)( 26, 89)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 96)( 32, 95)
( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,110)( 42,109)( 43,112)( 44,111)( 45,106)( 46,105)( 47,108)( 48,107)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,126)( 58,125)( 59,128)( 60,127)( 61,122)( 62,121)( 63,124)( 64,123)
(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)(136,200)
(137,202)(138,201)(139,204)(140,203)(141,206)(142,205)(143,208)(144,207)
(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)
(153,218)(154,217)(155,220)(156,219)(157,222)(158,221)(159,224)(160,223)
(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)(168,228)
(169,238)(170,237)(171,240)(172,239)(173,234)(174,233)(175,236)(176,235)
(177,245)(178,246)(179,247)(180,248)(181,241)(182,242)(183,243)(184,244)
(185,254)(186,253)(187,256)(188,255)(189,250)(190,249)(191,252)(192,251)
(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)(264,328)
(265,330)(266,329)(267,332)(268,331)(269,334)(270,333)(271,336)(272,335)
(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)(280,344)
(281,346)(282,345)(283,348)(284,347)(285,350)(286,349)(287,352)(288,351)
(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)(296,356)
(297,366)(298,365)(299,368)(300,367)(301,362)(302,361)(303,364)(304,363)
(305,373)(306,374)(307,375)(308,376)(309,369)(310,370)(311,371)(312,372)
(313,382)(314,381)(315,384)(316,383)(317,378)(318,377)(319,380)(320,379)
(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)(392,456)
(393,458)(394,457)(395,460)(396,459)(397,462)(398,461)(399,464)(400,463)
(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)(408,472)
(409,474)(410,473)(411,476)(412,475)(413,478)(414,477)(415,480)(416,479)
(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)(424,484)
(425,494)(426,493)(427,496)(428,495)(429,490)(430,489)(431,492)(432,491)
(433,501)(434,502)(435,503)(436,504)(437,497)(438,498)(439,499)(440,500)
(441,510)(442,509)(443,512)(444,511)(445,506)(446,505)(447,508)(448,507);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(512)!( 1,129)( 2,130)( 3,132)( 4,131)( 5,134)( 6,133)( 7,135)
( 8,136)( 9,137)( 10,138)( 11,140)( 12,139)( 13,142)( 14,141)( 15,143)
( 16,144)( 17,151)( 18,152)( 19,150)( 20,149)( 21,148)( 22,147)( 23,145)
( 24,146)( 25,159)( 26,160)( 27,158)( 28,157)( 29,156)( 30,155)( 31,153)
( 32,154)( 33,169)( 34,170)( 35,172)( 36,171)( 37,174)( 38,173)( 39,175)
( 40,176)( 41,161)( 42,162)( 43,164)( 44,163)( 45,166)( 46,165)( 47,167)
( 48,168)( 49,191)( 50,192)( 51,190)( 52,189)( 53,188)( 54,187)( 55,185)
( 56,186)( 57,183)( 58,184)( 59,182)( 60,181)( 61,180)( 62,179)( 63,177)
( 64,178)( 65,193)( 66,194)( 67,196)( 68,195)( 69,198)( 70,197)( 71,199)
( 72,200)( 73,201)( 74,202)( 75,204)( 76,203)( 77,206)( 78,205)( 79,207)
( 80,208)( 81,215)( 82,216)( 83,214)( 84,213)( 85,212)( 86,211)( 87,209)
( 88,210)( 89,223)( 90,224)( 91,222)( 92,221)( 93,220)( 94,219)( 95,217)
( 96,218)( 97,233)( 98,234)( 99,236)(100,235)(101,238)(102,237)(103,239)
(104,240)(105,225)(106,226)(107,228)(108,227)(109,230)(110,229)(111,231)
(112,232)(113,255)(114,256)(115,254)(116,253)(117,252)(118,251)(119,249)
(120,250)(121,247)(122,248)(123,246)(124,245)(125,244)(126,243)(127,241)
(128,242)(257,385)(258,386)(259,388)(260,387)(261,390)(262,389)(263,391)
(264,392)(265,393)(266,394)(267,396)(268,395)(269,398)(270,397)(271,399)
(272,400)(273,407)(274,408)(275,406)(276,405)(277,404)(278,403)(279,401)
(280,402)(281,415)(282,416)(283,414)(284,413)(285,412)(286,411)(287,409)
(288,410)(289,425)(290,426)(291,428)(292,427)(293,430)(294,429)(295,431)
(296,432)(297,417)(298,418)(299,420)(300,419)(301,422)(302,421)(303,423)
(304,424)(305,447)(306,448)(307,446)(308,445)(309,444)(310,443)(311,441)
(312,442)(313,439)(314,440)(315,438)(316,437)(317,436)(318,435)(319,433)
(320,434)(321,449)(322,450)(323,452)(324,451)(325,454)(326,453)(327,455)
(328,456)(329,457)(330,458)(331,460)(332,459)(333,462)(334,461)(335,463)
(336,464)(337,471)(338,472)(339,470)(340,469)(341,468)(342,467)(343,465)
(344,466)(345,479)(346,480)(347,478)(348,477)(349,476)(350,475)(351,473)
(352,474)(353,489)(354,490)(355,492)(356,491)(357,494)(358,493)(359,495)
(360,496)(361,481)(362,482)(363,484)(364,483)(365,486)(366,485)(367,487)
(368,488)(369,511)(370,512)(371,510)(372,509)(373,508)(374,507)(375,505)
(376,506)(377,503)(378,504)(379,502)(380,501)(381,500)(382,499)(383,497)
(384,498);
s1 := Sym(512)!( 1,257)( 2,258)( 3,259)( 4,260)( 5,263)( 6,264)( 7,261)
( 8,262)( 9,265)( 10,266)( 11,267)( 12,268)( 13,271)( 14,272)( 15,269)
( 16,270)( 17,273)( 18,274)( 19,275)( 20,276)( 21,279)( 22,280)( 23,277)
( 24,278)( 25,281)( 26,282)( 27,283)( 28,284)( 29,287)( 30,288)( 31,285)
( 32,286)( 33,297)( 34,298)( 35,299)( 36,300)( 37,303)( 38,304)( 39,301)
( 40,302)( 41,289)( 42,290)( 43,291)( 44,292)( 45,295)( 46,296)( 47,293)
( 48,294)( 49,313)( 50,314)( 51,315)( 52,316)( 53,319)( 54,320)( 55,317)
( 56,318)( 57,305)( 58,306)( 59,307)( 60,308)( 61,311)( 62,312)( 63,309)
( 64,310)( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)
( 72,342)( 73,345)( 74,346)( 75,347)( 76,348)( 77,351)( 78,352)( 79,349)
( 80,350)( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)
( 88,326)( 89,329)( 90,330)( 91,331)( 92,332)( 93,335)( 94,336)( 95,333)
( 96,334)( 97,380)( 98,379)( 99,378)(100,377)(101,382)(102,381)(103,384)
(104,383)(105,372)(106,371)(107,370)(108,369)(109,374)(110,373)(111,376)
(112,375)(113,364)(114,363)(115,362)(116,361)(117,366)(118,365)(119,368)
(120,367)(121,356)(122,355)(123,354)(124,353)(125,358)(126,357)(127,360)
(128,359)(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)
(136,422)(137,425)(138,426)(139,427)(140,428)(141,431)(142,432)(143,429)
(144,430)(145,433)(146,434)(147,435)(148,436)(149,439)(150,440)(151,437)
(152,438)(153,441)(154,442)(155,443)(156,444)(157,447)(158,448)(159,445)
(160,446)(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)
(168,390)(169,393)(170,394)(171,395)(172,396)(173,399)(174,400)(175,397)
(176,398)(177,401)(178,402)(179,403)(180,404)(181,407)(182,408)(183,405)
(184,406)(185,409)(186,410)(187,411)(188,412)(189,415)(190,416)(191,413)
(192,414)(193,501)(194,502)(195,503)(196,504)(197,499)(198,500)(199,497)
(200,498)(201,509)(202,510)(203,511)(204,512)(205,507)(206,508)(207,505)
(208,506)(209,485)(210,486)(211,487)(212,488)(213,483)(214,484)(215,481)
(216,482)(217,493)(218,494)(219,495)(220,496)(221,491)(222,492)(223,489)
(224,490)(225,471)(226,472)(227,469)(228,470)(229,465)(230,466)(231,467)
(232,468)(233,479)(234,480)(235,477)(236,478)(237,473)(238,474)(239,475)
(240,476)(241,455)(242,456)(243,453)(244,454)(245,449)(246,450)(247,451)
(248,452)(249,463)(250,464)(251,461)(252,462)(253,457)(254,458)(255,459)
(256,460);
s2 := Sym(512)!( 1, 65)( 2, 66)( 3, 67)( 4, 68)( 5, 69)( 6, 70)( 7, 71)
( 8, 72)( 9, 74)( 10, 73)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 80)
( 16, 79)( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)
( 24, 88)( 25, 90)( 26, 89)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 96)
( 32, 95)( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)
( 40,100)( 41,110)( 42,109)( 43,112)( 44,111)( 45,106)( 46,105)( 47,108)
( 48,107)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)
( 56,116)( 57,126)( 58,125)( 59,128)( 60,127)( 61,122)( 62,121)( 63,124)
( 64,123)(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)
(136,200)(137,202)(138,201)(139,204)(140,203)(141,206)(142,205)(143,208)
(144,207)(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)
(152,216)(153,218)(154,217)(155,220)(156,219)(157,222)(158,221)(159,224)
(160,223)(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)
(168,228)(169,238)(170,237)(171,240)(172,239)(173,234)(174,233)(175,236)
(176,235)(177,245)(178,246)(179,247)(180,248)(181,241)(182,242)(183,243)
(184,244)(185,254)(186,253)(187,256)(188,255)(189,250)(190,249)(191,252)
(192,251)(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)
(264,328)(265,330)(266,329)(267,332)(268,331)(269,334)(270,333)(271,336)
(272,335)(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)
(280,344)(281,346)(282,345)(283,348)(284,347)(285,350)(286,349)(287,352)
(288,351)(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)
(296,356)(297,366)(298,365)(299,368)(300,367)(301,362)(302,361)(303,364)
(304,363)(305,373)(306,374)(307,375)(308,376)(309,369)(310,370)(311,371)
(312,372)(313,382)(314,381)(315,384)(316,383)(317,378)(318,377)(319,380)
(320,379)(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)
(392,456)(393,458)(394,457)(395,460)(396,459)(397,462)(398,461)(399,464)
(400,463)(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)
(408,472)(409,474)(410,473)(411,476)(412,475)(413,478)(414,477)(415,480)
(416,479)(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)
(424,484)(425,494)(426,493)(427,496)(428,495)(429,490)(430,489)(431,492)
(432,491)(433,501)(434,502)(435,503)(436,504)(437,497)(438,498)(439,499)
(440,500)(441,510)(442,509)(443,512)(444,511)(445,506)(446,505)(447,508)
(448,507);
poly := sub<Sym(512)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope