Polytope of Type {8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4}*512b
if this polytope has a name.
Group : SmallGroup(512,58338)
Rank : 3
Schlafli Type : {8,4}
Number of vertices, edges, etc : 64, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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, {4,4}*256, {8,4}*256c
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)( 5,133)( 6,134)( 7,136)( 8,135)
( 9,137)( 10,138)( 11,140)( 12,139)( 13,141)( 14,142)( 15,144)( 16,143)
( 17,152)( 18,151)( 19,149)( 20,150)( 21,147)( 22,148)( 23,146)( 24,145)
( 25,160)( 26,159)( 27,157)( 28,158)( 29,155)( 30,156)( 31,154)( 32,153)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,173)( 38,174)( 39,176)( 40,175)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)( 48,167)
( 49,192)( 50,191)( 51,189)( 52,190)( 53,187)( 54,188)( 55,186)( 56,185)
( 57,184)( 58,183)( 59,181)( 60,182)( 61,179)( 62,180)( 63,178)( 64,177)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)( 72,199)
( 73,201)( 74,202)( 75,204)( 76,203)( 77,205)( 78,206)( 79,208)( 80,207)
( 81,216)( 82,215)( 83,213)( 84,214)( 85,211)( 86,212)( 87,210)( 88,209)
( 89,224)( 90,223)( 91,221)( 92,222)( 93,219)( 94,220)( 95,218)( 96,217)
( 97,233)( 98,234)( 99,236)(100,235)(101,237)(102,238)(103,240)(104,239)
(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)(112,231)
(113,256)(114,255)(115,253)(116,254)(117,251)(118,252)(119,250)(120,249)
(121,248)(122,247)(123,245)(124,246)(125,243)(126,244)(127,242)(128,241)
(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)(264,391)
(265,393)(266,394)(267,396)(268,395)(269,397)(270,398)(271,400)(272,399)
(273,408)(274,407)(275,405)(276,406)(277,403)(278,404)(279,402)(280,401)
(281,416)(282,415)(283,413)(284,414)(285,411)(286,412)(287,410)(288,409)
(289,425)(290,426)(291,428)(292,427)(293,429)(294,430)(295,432)(296,431)
(297,417)(298,418)(299,420)(300,419)(301,421)(302,422)(303,424)(304,423)
(305,448)(306,447)(307,445)(308,446)(309,443)(310,444)(311,442)(312,441)
(313,440)(314,439)(315,437)(316,438)(317,435)(318,436)(319,434)(320,433)
(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)(328,455)
(329,457)(330,458)(331,460)(332,459)(333,461)(334,462)(335,464)(336,463)
(337,472)(338,471)(339,469)(340,470)(341,467)(342,468)(343,466)(344,465)
(345,480)(346,479)(347,477)(348,478)(349,475)(350,476)(351,474)(352,473)
(353,489)(354,490)(355,492)(356,491)(357,493)(358,494)(359,496)(360,495)
(361,481)(362,482)(363,484)(364,483)(365,485)(366,486)(367,488)(368,487)
(369,512)(370,511)(371,509)(372,510)(373,507)(374,508)(375,506)(376,505)
(377,504)(378,503)(379,501)(380,502)(381,499)(382,500)(383,498)(384,497);;
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)
(193,502)(194,501)(195,504)(196,503)(197,499)(198,500)(199,497)(200,498)
(201,510)(202,509)(203,512)(204,511)(205,507)(206,508)(207,505)(208,506)
(209,485)(210,486)(211,487)(212,488)(213,484)(214,483)(215,482)(216,481)
(217,493)(218,494)(219,495)(220,496)(221,492)(222,491)(223,490)(224,489)
(225,472)(226,471)(227,470)(228,469)(229,465)(230,466)(231,467)(232,468)
(233,480)(234,479)(235,478)(236,477)(237,473)(238,474)(239,475)(240,476)
(241,455)(242,456)(243,453)(244,454)(245,450)(246,449)(247,452)(248,451)
(249,463)(250,464)(251,461)(252,462)(253,458)(254,457)(255,460)(256,459);;
s2 := ( 1, 65)( 2, 66)( 3, 68)( 4, 67)( 5, 70)( 6, 69)( 7, 71)( 8, 72)
( 9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 79)( 16, 80)
( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 87)( 24, 88)
( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 95)( 32, 96)
( 33,101)( 34,102)( 35,104)( 36,103)( 37, 97)( 38, 98)( 39,100)( 40, 99)
( 41,109)( 42,110)( 43,112)( 44,111)( 45,105)( 46,106)( 47,108)( 48,107)
( 49,118)( 50,117)( 51,119)( 52,120)( 53,114)( 54,113)( 55,115)( 56,116)
( 57,126)( 58,125)( 59,127)( 60,128)( 61,122)( 62,121)( 63,123)( 64,124)
(129,193)(130,194)(131,196)(132,195)(133,198)(134,197)(135,199)(136,200)
(137,201)(138,202)(139,204)(140,203)(141,206)(142,205)(143,207)(144,208)
(145,209)(146,210)(147,212)(148,211)(149,214)(150,213)(151,215)(152,216)
(153,217)(154,218)(155,220)(156,219)(157,222)(158,221)(159,223)(160,224)
(161,229)(162,230)(163,232)(164,231)(165,225)(166,226)(167,228)(168,227)
(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)(176,235)
(177,246)(178,245)(179,247)(180,248)(181,242)(182,241)(183,243)(184,244)
(185,254)(186,253)(187,255)(188,256)(189,250)(190,249)(191,251)(192,252)
(257,321)(258,322)(259,324)(260,323)(261,326)(262,325)(263,327)(264,328)
(265,329)(266,330)(267,332)(268,331)(269,334)(270,333)(271,335)(272,336)
(273,337)(274,338)(275,340)(276,339)(277,342)(278,341)(279,343)(280,344)
(281,345)(282,346)(283,348)(284,347)(285,350)(286,349)(287,351)(288,352)
(289,357)(290,358)(291,360)(292,359)(293,353)(294,354)(295,356)(296,355)
(297,365)(298,366)(299,368)(300,367)(301,361)(302,362)(303,364)(304,363)
(305,374)(306,373)(307,375)(308,376)(309,370)(310,369)(311,371)(312,372)
(313,382)(314,381)(315,383)(316,384)(317,378)(318,377)(319,379)(320,380)
(385,449)(386,450)(387,452)(388,451)(389,454)(390,453)(391,455)(392,456)
(393,457)(394,458)(395,460)(396,459)(397,462)(398,461)(399,463)(400,464)
(401,465)(402,466)(403,468)(404,467)(405,470)(406,469)(407,471)(408,472)
(409,473)(410,474)(411,476)(412,475)(413,478)(414,477)(415,479)(416,480)
(417,485)(418,486)(419,488)(420,487)(421,481)(422,482)(423,484)(424,483)
(425,493)(426,494)(427,496)(428,495)(429,489)(430,490)(431,492)(432,491)
(433,502)(434,501)(435,503)(436,504)(437,498)(438,497)(439,499)(440,500)
(441,510)(442,509)(443,511)(444,512)(445,506)(446,505)(447,507)(448,508);;
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(512)!( 1,129)( 2,130)( 3,132)( 4,131)( 5,133)( 6,134)( 7,136)
( 8,135)( 9,137)( 10,138)( 11,140)( 12,139)( 13,141)( 14,142)( 15,144)
( 16,143)( 17,152)( 18,151)( 19,149)( 20,150)( 21,147)( 22,148)( 23,146)
( 24,145)( 25,160)( 26,159)( 27,157)( 28,158)( 29,155)( 30,156)( 31,154)
( 32,153)( 33,169)( 34,170)( 35,172)( 36,171)( 37,173)( 38,174)( 39,176)
( 40,175)( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)
( 48,167)( 49,192)( 50,191)( 51,189)( 52,190)( 53,187)( 54,188)( 55,186)
( 56,185)( 57,184)( 58,183)( 59,181)( 60,182)( 61,179)( 62,180)( 63,178)
( 64,177)( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)
( 72,199)( 73,201)( 74,202)( 75,204)( 76,203)( 77,205)( 78,206)( 79,208)
( 80,207)( 81,216)( 82,215)( 83,213)( 84,214)( 85,211)( 86,212)( 87,210)
( 88,209)( 89,224)( 90,223)( 91,221)( 92,222)( 93,219)( 94,220)( 95,218)
( 96,217)( 97,233)( 98,234)( 99,236)(100,235)(101,237)(102,238)(103,240)
(104,239)(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)
(112,231)(113,256)(114,255)(115,253)(116,254)(117,251)(118,252)(119,250)
(120,249)(121,248)(122,247)(123,245)(124,246)(125,243)(126,244)(127,242)
(128,241)(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)
(264,391)(265,393)(266,394)(267,396)(268,395)(269,397)(270,398)(271,400)
(272,399)(273,408)(274,407)(275,405)(276,406)(277,403)(278,404)(279,402)
(280,401)(281,416)(282,415)(283,413)(284,414)(285,411)(286,412)(287,410)
(288,409)(289,425)(290,426)(291,428)(292,427)(293,429)(294,430)(295,432)
(296,431)(297,417)(298,418)(299,420)(300,419)(301,421)(302,422)(303,424)
(304,423)(305,448)(306,447)(307,445)(308,446)(309,443)(310,444)(311,442)
(312,441)(313,440)(314,439)(315,437)(316,438)(317,435)(318,436)(319,434)
(320,433)(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)
(328,455)(329,457)(330,458)(331,460)(332,459)(333,461)(334,462)(335,464)
(336,463)(337,472)(338,471)(339,469)(340,470)(341,467)(342,468)(343,466)
(344,465)(345,480)(346,479)(347,477)(348,478)(349,475)(350,476)(351,474)
(352,473)(353,489)(354,490)(355,492)(356,491)(357,493)(358,494)(359,496)
(360,495)(361,481)(362,482)(363,484)(364,483)(365,485)(366,486)(367,488)
(368,487)(369,512)(370,511)(371,509)(372,510)(373,507)(374,508)(375,506)
(376,505)(377,504)(378,503)(379,501)(380,502)(381,499)(382,500)(383,498)
(384,497);
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)(193,502)(194,501)(195,504)(196,503)(197,499)(198,500)(199,497)
(200,498)(201,510)(202,509)(203,512)(204,511)(205,507)(206,508)(207,505)
(208,506)(209,485)(210,486)(211,487)(212,488)(213,484)(214,483)(215,482)
(216,481)(217,493)(218,494)(219,495)(220,496)(221,492)(222,491)(223,490)
(224,489)(225,472)(226,471)(227,470)(228,469)(229,465)(230,466)(231,467)
(232,468)(233,480)(234,479)(235,478)(236,477)(237,473)(238,474)(239,475)
(240,476)(241,455)(242,456)(243,453)(244,454)(245,450)(246,449)(247,452)
(248,451)(249,463)(250,464)(251,461)(252,462)(253,458)(254,457)(255,460)
(256,459);
s2 := Sym(512)!( 1, 65)( 2, 66)( 3, 68)( 4, 67)( 5, 70)( 6, 69)( 7, 71)
( 8, 72)( 9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 79)
( 16, 80)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 87)
( 24, 88)( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 95)
( 32, 96)( 33,101)( 34,102)( 35,104)( 36,103)( 37, 97)( 38, 98)( 39,100)
( 40, 99)( 41,109)( 42,110)( 43,112)( 44,111)( 45,105)( 46,106)( 47,108)
( 48,107)( 49,118)( 50,117)( 51,119)( 52,120)( 53,114)( 54,113)( 55,115)
( 56,116)( 57,126)( 58,125)( 59,127)( 60,128)( 61,122)( 62,121)( 63,123)
( 64,124)(129,193)(130,194)(131,196)(132,195)(133,198)(134,197)(135,199)
(136,200)(137,201)(138,202)(139,204)(140,203)(141,206)(142,205)(143,207)
(144,208)(145,209)(146,210)(147,212)(148,211)(149,214)(150,213)(151,215)
(152,216)(153,217)(154,218)(155,220)(156,219)(157,222)(158,221)(159,223)
(160,224)(161,229)(162,230)(163,232)(164,231)(165,225)(166,226)(167,228)
(168,227)(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)
(176,235)(177,246)(178,245)(179,247)(180,248)(181,242)(182,241)(183,243)
(184,244)(185,254)(186,253)(187,255)(188,256)(189,250)(190,249)(191,251)
(192,252)(257,321)(258,322)(259,324)(260,323)(261,326)(262,325)(263,327)
(264,328)(265,329)(266,330)(267,332)(268,331)(269,334)(270,333)(271,335)
(272,336)(273,337)(274,338)(275,340)(276,339)(277,342)(278,341)(279,343)
(280,344)(281,345)(282,346)(283,348)(284,347)(285,350)(286,349)(287,351)
(288,352)(289,357)(290,358)(291,360)(292,359)(293,353)(294,354)(295,356)
(296,355)(297,365)(298,366)(299,368)(300,367)(301,361)(302,362)(303,364)
(304,363)(305,374)(306,373)(307,375)(308,376)(309,370)(310,369)(311,371)
(312,372)(313,382)(314,381)(315,383)(316,384)(317,378)(318,377)(319,379)
(320,380)(385,449)(386,450)(387,452)(388,451)(389,454)(390,453)(391,455)
(392,456)(393,457)(394,458)(395,460)(396,459)(397,462)(398,461)(399,463)
(400,464)(401,465)(402,466)(403,468)(404,467)(405,470)(406,469)(407,471)
(408,472)(409,473)(410,474)(411,476)(412,475)(413,478)(414,477)(415,479)
(416,480)(417,485)(418,486)(419,488)(420,487)(421,481)(422,482)(423,484)
(424,483)(425,493)(426,494)(427,496)(428,495)(429,489)(430,490)(431,492)
(432,491)(433,502)(434,501)(435,503)(436,504)(437,498)(438,497)(439,499)
(440,500)(441,510)(442,509)(443,511)(444,512)(445,506)(446,505)(447,507)
(448,508);
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope