Polytope of Type {8,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*512g
if this polytope has a name.
Group : SmallGroup(512,58328)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 32, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,8}*256b
4-fold quotients : {4,8}*128b, {8,4}*128b
8-fold quotients : {4,4}*64
16-fold quotients : {4,4}*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)
( 17,274)( 18,273)( 19,276)( 20,275)( 21,280)( 22,279)( 23,278)( 24,277)
( 25,282)( 26,281)( 27,284)( 28,283)( 29,288)( 30,287)( 31,286)( 32,285)
( 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)
( 49,314)( 50,313)( 51,316)( 52,315)( 53,320)( 54,319)( 55,318)( 56,317)
( 57,306)( 58,305)( 59,308)( 60,307)( 61,312)( 62,311)( 63,310)( 64,309)
( 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)
(145,434)(146,433)(147,436)(148,435)(149,440)(150,439)(151,438)(152,437)
(153,442)(154,441)(155,444)(156,443)(157,448)(158,447)(159,446)(160,445)
(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)
(177,402)(178,401)(179,404)(180,403)(181,408)(182,407)(183,406)(184,405)
(185,410)(186,409)(187,412)(188,411)(189,416)(190,415)(191,414)(192,413)
(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 := ( 1,193)( 2,194)( 3,196)( 4,195)( 5,198)( 6,197)( 7,199)( 8,200)
( 9,202)( 10,201)( 11,203)( 12,204)( 13,205)( 14,206)( 15,208)( 16,207)
( 17,216)( 18,215)( 19,213)( 20,214)( 21,211)( 22,212)( 23,210)( 24,209)
( 25,223)( 26,224)( 27,222)( 28,221)( 29,220)( 30,219)( 31,217)( 32,218)
( 33,238)( 34,237)( 35,239)( 36,240)( 37,233)( 38,234)( 39,236)( 40,235)
( 41,229)( 42,230)( 43,232)( 44,231)( 45,226)( 46,225)( 47,227)( 48,228)
( 49,251)( 50,252)( 51,250)( 52,249)( 53,256)( 54,255)( 55,253)( 56,254)
( 57,244)( 58,243)( 59,241)( 60,242)( 61,247)( 62,248)( 63,246)( 64,245)
( 65,129)( 66,130)( 67,132)( 68,131)( 69,134)( 70,133)( 71,135)( 72,136)
( 73,138)( 74,137)( 75,139)( 76,140)( 77,141)( 78,142)( 79,144)( 80,143)
( 81,152)( 82,151)( 83,149)( 84,150)( 85,147)( 86,148)( 87,146)( 88,145)
( 89,159)( 90,160)( 91,158)( 92,157)( 93,156)( 94,155)( 95,153)( 96,154)
( 97,174)( 98,173)( 99,175)(100,176)(101,169)(102,170)(103,172)(104,171)
(105,165)(106,166)(107,168)(108,167)(109,162)(110,161)(111,163)(112,164)
(113,187)(114,188)(115,186)(116,185)(117,192)(118,191)(119,189)(120,190)
(121,180)(122,179)(123,177)(124,178)(125,183)(126,184)(127,182)(128,181)
(257,449)(258,450)(259,452)(260,451)(261,454)(262,453)(263,455)(264,456)
(265,458)(266,457)(267,459)(268,460)(269,461)(270,462)(271,464)(272,463)
(273,472)(274,471)(275,469)(276,470)(277,467)(278,468)(279,466)(280,465)
(281,479)(282,480)(283,478)(284,477)(285,476)(286,475)(287,473)(288,474)
(289,494)(290,493)(291,495)(292,496)(293,489)(294,490)(295,492)(296,491)
(297,485)(298,486)(299,488)(300,487)(301,482)(302,481)(303,483)(304,484)
(305,507)(306,508)(307,506)(308,505)(309,512)(310,511)(311,509)(312,510)
(313,500)(314,499)(315,497)(316,498)(317,503)(318,504)(319,502)(320,501)
(321,385)(322,386)(323,388)(324,387)(325,390)(326,389)(327,391)(328,392)
(329,394)(330,393)(331,395)(332,396)(333,397)(334,398)(335,400)(336,399)
(337,408)(338,407)(339,405)(340,406)(341,403)(342,404)(343,402)(344,401)
(345,415)(346,416)(347,414)(348,413)(349,412)(350,411)(351,409)(352,410)
(353,430)(354,429)(355,431)(356,432)(357,425)(358,426)(359,428)(360,427)
(361,421)(362,422)(363,424)(364,423)(365,418)(366,417)(367,419)(368,420)
(369,443)(370,444)(371,442)(372,441)(373,448)(374,447)(375,445)(376,446)
(377,436)(378,435)(379,433)(380,434)(381,439)(382,440)(383,438)(384,437);;
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, 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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*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)( 17,274)( 18,273)( 19,276)( 20,275)( 21,280)( 22,279)( 23,278)
( 24,277)( 25,282)( 26,281)( 27,284)( 28,283)( 29,288)( 30,287)( 31,286)
( 32,285)( 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)( 49,314)( 50,313)( 51,316)( 52,315)( 53,320)( 54,319)( 55,318)
( 56,317)( 57,306)( 58,305)( 59,308)( 60,307)( 61,312)( 62,311)( 63,310)
( 64,309)( 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)(145,434)(146,433)(147,436)(148,435)(149,440)(150,439)(151,438)
(152,437)(153,442)(154,441)(155,444)(156,443)(157,448)(158,447)(159,446)
(160,445)(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)(177,402)(178,401)(179,404)(180,403)(181,408)(182,407)(183,406)
(184,405)(185,410)(186,409)(187,412)(188,411)(189,416)(190,415)(191,414)
(192,413)(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)!( 1,193)( 2,194)( 3,196)( 4,195)( 5,198)( 6,197)( 7,199)
( 8,200)( 9,202)( 10,201)( 11,203)( 12,204)( 13,205)( 14,206)( 15,208)
( 16,207)( 17,216)( 18,215)( 19,213)( 20,214)( 21,211)( 22,212)( 23,210)
( 24,209)( 25,223)( 26,224)( 27,222)( 28,221)( 29,220)( 30,219)( 31,217)
( 32,218)( 33,238)( 34,237)( 35,239)( 36,240)( 37,233)( 38,234)( 39,236)
( 40,235)( 41,229)( 42,230)( 43,232)( 44,231)( 45,226)( 46,225)( 47,227)
( 48,228)( 49,251)( 50,252)( 51,250)( 52,249)( 53,256)( 54,255)( 55,253)
( 56,254)( 57,244)( 58,243)( 59,241)( 60,242)( 61,247)( 62,248)( 63,246)
( 64,245)( 65,129)( 66,130)( 67,132)( 68,131)( 69,134)( 70,133)( 71,135)
( 72,136)( 73,138)( 74,137)( 75,139)( 76,140)( 77,141)( 78,142)( 79,144)
( 80,143)( 81,152)( 82,151)( 83,149)( 84,150)( 85,147)( 86,148)( 87,146)
( 88,145)( 89,159)( 90,160)( 91,158)( 92,157)( 93,156)( 94,155)( 95,153)
( 96,154)( 97,174)( 98,173)( 99,175)(100,176)(101,169)(102,170)(103,172)
(104,171)(105,165)(106,166)(107,168)(108,167)(109,162)(110,161)(111,163)
(112,164)(113,187)(114,188)(115,186)(116,185)(117,192)(118,191)(119,189)
(120,190)(121,180)(122,179)(123,177)(124,178)(125,183)(126,184)(127,182)
(128,181)(257,449)(258,450)(259,452)(260,451)(261,454)(262,453)(263,455)
(264,456)(265,458)(266,457)(267,459)(268,460)(269,461)(270,462)(271,464)
(272,463)(273,472)(274,471)(275,469)(276,470)(277,467)(278,468)(279,466)
(280,465)(281,479)(282,480)(283,478)(284,477)(285,476)(286,475)(287,473)
(288,474)(289,494)(290,493)(291,495)(292,496)(293,489)(294,490)(295,492)
(296,491)(297,485)(298,486)(299,488)(300,487)(301,482)(302,481)(303,483)
(304,484)(305,507)(306,508)(307,506)(308,505)(309,512)(310,511)(311,509)
(312,510)(313,500)(314,499)(315,497)(316,498)(317,503)(318,504)(319,502)
(320,501)(321,385)(322,386)(323,388)(324,387)(325,390)(326,389)(327,391)
(328,392)(329,394)(330,393)(331,395)(332,396)(333,397)(334,398)(335,400)
(336,399)(337,408)(338,407)(339,405)(340,406)(341,403)(342,404)(343,402)
(344,401)(345,415)(346,416)(347,414)(348,413)(349,412)(350,411)(351,409)
(352,410)(353,430)(354,429)(355,431)(356,432)(357,425)(358,426)(359,428)
(360,427)(361,421)(362,422)(363,424)(364,423)(365,418)(366,417)(367,419)
(368,420)(369,443)(370,444)(371,442)(372,441)(373,448)(374,447)(375,445)
(376,446)(377,436)(378,435)(379,433)(380,434)(381,439)(382,440)(383,438)
(384,437);
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, 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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1 >;
References : None.
to this polytope