Polytope of Type {4,64}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,64}*512a
Also Known As : {4,64|2}. if this polytope has another name.
Group : SmallGroup(512,60797)
Rank : 3
Schlafli Type : {4,64}
Number of vertices, edges, etc : 4, 128, 64
Order of s0s1s2 : 64
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {4,32}*256a, {2,64}*256
4-fold quotients : {4,16}*128a, {2,32}*128
8-fold quotients : {4,8}*64a, {2,16}*64
16-fold quotients : {4,4}*32, {2,8}*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, 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, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)( 24, 88)
( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)( 32, 96)
( 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)
( 49,113)( 50,114)( 51,115)( 52,116)( 53,117)( 54,118)( 55,119)( 56,120)
( 57,121)( 58,122)( 59,123)( 60,124)( 61,125)( 62,126)( 63,127)( 64,128)
(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)
(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)
(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)(160,224)
(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)
(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)(184,248)
(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)(192,256)
(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)
(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)(280,344)
(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)(288,352)
(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)
(305,369)(306,370)(307,371)(308,372)(309,373)(310,374)(311,375)(312,376)
(313,377)(314,378)(315,379)(316,380)(317,381)(318,382)(319,383)(320,384)
(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)
(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)(408,472)
(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)(416,480)
(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)
(433,497)(434,498)(435,499)(436,500)(437,501)(438,502)(439,503)(440,504)
(441,505)(442,506)(443,507)(444,508)(445,509)(446,510)(447,511)(448,512);;
s1 := ( 1,257)( 2,258)( 3,260)( 4,259)( 5,263)( 6,264)( 7,261)( 8,262)
( 9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)( 16,267)
( 17,273)( 18,274)( 19,276)( 20,275)( 21,279)( 22,280)( 23,277)( 24,278)
( 25,285)( 26,286)( 27,288)( 28,287)( 29,281)( 30,282)( 31,284)( 32,283)
( 33,297)( 34,298)( 35,300)( 36,299)( 37,303)( 38,304)( 39,301)( 40,302)
( 41,289)( 42,290)( 43,292)( 44,291)( 45,295)( 46,296)( 47,293)( 48,294)
( 49,313)( 50,314)( 51,316)( 52,315)( 53,319)( 54,320)( 55,317)( 56,318)
( 57,305)( 58,306)( 59,308)( 60,307)( 61,311)( 62,312)( 63,309)( 64,310)
( 65,337)( 66,338)( 67,340)( 68,339)( 69,343)( 70,344)( 71,341)( 72,342)
( 73,349)( 74,350)( 75,352)( 76,351)( 77,345)( 78,346)( 79,348)( 80,347)
( 81,321)( 82,322)( 83,324)( 84,323)( 85,327)( 86,328)( 87,325)( 88,326)
( 89,333)( 90,334)( 91,336)( 92,335)( 93,329)( 94,330)( 95,332)( 96,331)
( 97,377)( 98,378)( 99,380)(100,379)(101,383)(102,384)(103,381)(104,382)
(105,369)(106,370)(107,372)(108,371)(109,375)(110,376)(111,373)(112,374)
(113,361)(114,362)(115,364)(116,363)(117,367)(118,368)(119,365)(120,366)
(121,353)(122,354)(123,356)(124,355)(125,359)(126,360)(127,357)(128,358)
(129,417)(130,418)(131,420)(132,419)(133,423)(134,424)(135,421)(136,422)
(137,429)(138,430)(139,432)(140,431)(141,425)(142,426)(143,428)(144,427)
(145,433)(146,434)(147,436)(148,435)(149,439)(150,440)(151,437)(152,438)
(153,445)(154,446)(155,448)(156,447)(157,441)(158,442)(159,444)(160,443)
(161,385)(162,386)(163,388)(164,387)(165,391)(166,392)(167,389)(168,390)
(169,397)(170,398)(171,400)(172,399)(173,393)(174,394)(175,396)(176,395)
(177,401)(178,402)(179,404)(180,403)(181,407)(182,408)(183,405)(184,406)
(185,413)(186,414)(187,416)(188,415)(189,409)(190,410)(191,412)(192,411)
(193,497)(194,498)(195,500)(196,499)(197,503)(198,504)(199,501)(200,502)
(201,509)(202,510)(203,512)(204,511)(205,505)(206,506)(207,508)(208,507)
(209,481)(210,482)(211,484)(212,483)(213,487)(214,488)(215,485)(216,486)
(217,493)(218,494)(219,496)(220,495)(221,489)(222,490)(223,492)(224,491)
(225,465)(226,466)(227,468)(228,467)(229,471)(230,472)(231,469)(232,470)
(233,477)(234,478)(235,480)(236,479)(237,473)(238,474)(239,476)(240,475)
(241,449)(242,450)(243,452)(244,451)(245,455)(246,456)(247,453)(248,454)
(249,461)(250,462)(251,464)(252,463)(253,457)(254,458)(255,460)(256,459);;
s2 := ( 1,129)( 2,130)( 3,132)( 4,131)( 5,135)( 6,136)( 7,133)( 8,134)
( 9,141)( 10,142)( 11,144)( 12,143)( 13,137)( 14,138)( 15,140)( 16,139)
( 17,145)( 18,146)( 19,148)( 20,147)( 21,151)( 22,152)( 23,149)( 24,150)
( 25,157)( 26,158)( 27,160)( 28,159)( 29,153)( 30,154)( 31,156)( 32,155)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,175)( 38,176)( 39,173)( 40,174)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,185)( 50,186)( 51,188)( 52,187)( 53,191)( 54,192)( 55,189)( 56,190)
( 57,177)( 58,178)( 59,180)( 60,179)( 61,183)( 62,184)( 63,181)( 64,182)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,199)( 70,200)( 71,197)( 72,198)
( 73,205)( 74,206)( 75,208)( 76,207)( 77,201)( 78,202)( 79,204)( 80,203)
( 81,209)( 82,210)( 83,212)( 84,211)( 85,215)( 86,216)( 87,213)( 88,214)
( 89,221)( 90,222)( 91,224)( 92,223)( 93,217)( 94,218)( 95,220)( 96,219)
( 97,233)( 98,234)( 99,236)(100,235)(101,239)(102,240)(103,237)(104,238)
(105,225)(106,226)(107,228)(108,227)(109,231)(110,232)(111,229)(112,230)
(113,249)(114,250)(115,252)(116,251)(117,255)(118,256)(119,253)(120,254)
(121,241)(122,242)(123,244)(124,243)(125,247)(126,248)(127,245)(128,246)
(257,385)(258,386)(259,388)(260,387)(261,391)(262,392)(263,389)(264,390)
(265,397)(266,398)(267,400)(268,399)(269,393)(270,394)(271,396)(272,395)
(273,401)(274,402)(275,404)(276,403)(277,407)(278,408)(279,405)(280,406)
(281,413)(282,414)(283,416)(284,415)(285,409)(286,410)(287,412)(288,411)
(289,425)(290,426)(291,428)(292,427)(293,431)(294,432)(295,429)(296,430)
(297,417)(298,418)(299,420)(300,419)(301,423)(302,424)(303,421)(304,422)
(305,441)(306,442)(307,444)(308,443)(309,447)(310,448)(311,445)(312,446)
(313,433)(314,434)(315,436)(316,435)(317,439)(318,440)(319,437)(320,438)
(321,449)(322,450)(323,452)(324,451)(325,455)(326,456)(327,453)(328,454)
(329,461)(330,462)(331,464)(332,463)(333,457)(334,458)(335,460)(336,459)
(337,465)(338,466)(339,468)(340,467)(341,471)(342,472)(343,469)(344,470)
(345,477)(346,478)(347,480)(348,479)(349,473)(350,474)(351,476)(352,475)
(353,489)(354,490)(355,492)(356,491)(357,495)(358,496)(359,493)(360,494)
(361,481)(362,482)(363,484)(364,483)(365,487)(366,488)(367,485)(368,486)
(369,505)(370,506)(371,508)(372,507)(373,511)(374,512)(375,509)(376,510)
(377,497)(378,498)(379,500)(380,499)(381,503)(382,504)(383,501)(384,502);;
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 87)
( 24, 88)( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)
( 32, 96)( 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)( 49,113)( 50,114)( 51,115)( 52,116)( 53,117)( 54,118)( 55,119)
( 56,120)( 57,121)( 58,122)( 59,123)( 60,124)( 61,125)( 62,126)( 63,127)
( 64,128)(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)(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)
(152,216)(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)
(160,224)(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)(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)
(184,248)(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)
(192,256)(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)(273,337)(274,338)(275,339)(276,340)(277,341)(278,342)(279,343)
(280,344)(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)
(288,352)(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)(305,369)(306,370)(307,371)(308,372)(309,373)(310,374)(311,375)
(312,376)(313,377)(314,378)(315,379)(316,380)(317,381)(318,382)(319,383)
(320,384)(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)(401,465)(402,466)(403,467)(404,468)(405,469)(406,470)(407,471)
(408,472)(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)
(416,480)(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)(433,497)(434,498)(435,499)(436,500)(437,501)(438,502)(439,503)
(440,504)(441,505)(442,506)(443,507)(444,508)(445,509)(446,510)(447,511)
(448,512);
s1 := Sym(512)!( 1,257)( 2,258)( 3,260)( 4,259)( 5,263)( 6,264)( 7,261)
( 8,262)( 9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)
( 16,267)( 17,273)( 18,274)( 19,276)( 20,275)( 21,279)( 22,280)( 23,277)
( 24,278)( 25,285)( 26,286)( 27,288)( 28,287)( 29,281)( 30,282)( 31,284)
( 32,283)( 33,297)( 34,298)( 35,300)( 36,299)( 37,303)( 38,304)( 39,301)
( 40,302)( 41,289)( 42,290)( 43,292)( 44,291)( 45,295)( 46,296)( 47,293)
( 48,294)( 49,313)( 50,314)( 51,316)( 52,315)( 53,319)( 54,320)( 55,317)
( 56,318)( 57,305)( 58,306)( 59,308)( 60,307)( 61,311)( 62,312)( 63,309)
( 64,310)( 65,337)( 66,338)( 67,340)( 68,339)( 69,343)( 70,344)( 71,341)
( 72,342)( 73,349)( 74,350)( 75,352)( 76,351)( 77,345)( 78,346)( 79,348)
( 80,347)( 81,321)( 82,322)( 83,324)( 84,323)( 85,327)( 86,328)( 87,325)
( 88,326)( 89,333)( 90,334)( 91,336)( 92,335)( 93,329)( 94,330)( 95,332)
( 96,331)( 97,377)( 98,378)( 99,380)(100,379)(101,383)(102,384)(103,381)
(104,382)(105,369)(106,370)(107,372)(108,371)(109,375)(110,376)(111,373)
(112,374)(113,361)(114,362)(115,364)(116,363)(117,367)(118,368)(119,365)
(120,366)(121,353)(122,354)(123,356)(124,355)(125,359)(126,360)(127,357)
(128,358)(129,417)(130,418)(131,420)(132,419)(133,423)(134,424)(135,421)
(136,422)(137,429)(138,430)(139,432)(140,431)(141,425)(142,426)(143,428)
(144,427)(145,433)(146,434)(147,436)(148,435)(149,439)(150,440)(151,437)
(152,438)(153,445)(154,446)(155,448)(156,447)(157,441)(158,442)(159,444)
(160,443)(161,385)(162,386)(163,388)(164,387)(165,391)(166,392)(167,389)
(168,390)(169,397)(170,398)(171,400)(172,399)(173,393)(174,394)(175,396)
(176,395)(177,401)(178,402)(179,404)(180,403)(181,407)(182,408)(183,405)
(184,406)(185,413)(186,414)(187,416)(188,415)(189,409)(190,410)(191,412)
(192,411)(193,497)(194,498)(195,500)(196,499)(197,503)(198,504)(199,501)
(200,502)(201,509)(202,510)(203,512)(204,511)(205,505)(206,506)(207,508)
(208,507)(209,481)(210,482)(211,484)(212,483)(213,487)(214,488)(215,485)
(216,486)(217,493)(218,494)(219,496)(220,495)(221,489)(222,490)(223,492)
(224,491)(225,465)(226,466)(227,468)(228,467)(229,471)(230,472)(231,469)
(232,470)(233,477)(234,478)(235,480)(236,479)(237,473)(238,474)(239,476)
(240,475)(241,449)(242,450)(243,452)(244,451)(245,455)(246,456)(247,453)
(248,454)(249,461)(250,462)(251,464)(252,463)(253,457)(254,458)(255,460)
(256,459);
s2 := Sym(512)!( 1,129)( 2,130)( 3,132)( 4,131)( 5,135)( 6,136)( 7,133)
( 8,134)( 9,141)( 10,142)( 11,144)( 12,143)( 13,137)( 14,138)( 15,140)
( 16,139)( 17,145)( 18,146)( 19,148)( 20,147)( 21,151)( 22,152)( 23,149)
( 24,150)( 25,157)( 26,158)( 27,160)( 28,159)( 29,153)( 30,154)( 31,156)
( 32,155)( 33,169)( 34,170)( 35,172)( 36,171)( 37,175)( 38,176)( 39,173)
( 40,174)( 41,161)( 42,162)( 43,164)( 44,163)( 45,167)( 46,168)( 47,165)
( 48,166)( 49,185)( 50,186)( 51,188)( 52,187)( 53,191)( 54,192)( 55,189)
( 56,190)( 57,177)( 58,178)( 59,180)( 60,179)( 61,183)( 62,184)( 63,181)
( 64,182)( 65,193)( 66,194)( 67,196)( 68,195)( 69,199)( 70,200)( 71,197)
( 72,198)( 73,205)( 74,206)( 75,208)( 76,207)( 77,201)( 78,202)( 79,204)
( 80,203)( 81,209)( 82,210)( 83,212)( 84,211)( 85,215)( 86,216)( 87,213)
( 88,214)( 89,221)( 90,222)( 91,224)( 92,223)( 93,217)( 94,218)( 95,220)
( 96,219)( 97,233)( 98,234)( 99,236)(100,235)(101,239)(102,240)(103,237)
(104,238)(105,225)(106,226)(107,228)(108,227)(109,231)(110,232)(111,229)
(112,230)(113,249)(114,250)(115,252)(116,251)(117,255)(118,256)(119,253)
(120,254)(121,241)(122,242)(123,244)(124,243)(125,247)(126,248)(127,245)
(128,246)(257,385)(258,386)(259,388)(260,387)(261,391)(262,392)(263,389)
(264,390)(265,397)(266,398)(267,400)(268,399)(269,393)(270,394)(271,396)
(272,395)(273,401)(274,402)(275,404)(276,403)(277,407)(278,408)(279,405)
(280,406)(281,413)(282,414)(283,416)(284,415)(285,409)(286,410)(287,412)
(288,411)(289,425)(290,426)(291,428)(292,427)(293,431)(294,432)(295,429)
(296,430)(297,417)(298,418)(299,420)(300,419)(301,423)(302,424)(303,421)
(304,422)(305,441)(306,442)(307,444)(308,443)(309,447)(310,448)(311,445)
(312,446)(313,433)(314,434)(315,436)(316,435)(317,439)(318,440)(319,437)
(320,438)(321,449)(322,450)(323,452)(324,451)(325,455)(326,456)(327,453)
(328,454)(329,461)(330,462)(331,464)(332,463)(333,457)(334,458)(335,460)
(336,459)(337,465)(338,466)(339,468)(340,467)(341,471)(342,472)(343,469)
(344,470)(345,477)(346,478)(347,480)(348,479)(349,473)(350,474)(351,476)
(352,475)(353,489)(354,490)(355,492)(356,491)(357,495)(358,496)(359,493)
(360,494)(361,481)(362,482)(363,484)(364,483)(365,487)(366,488)(367,485)
(368,486)(369,505)(370,506)(371,508)(372,507)(373,511)(374,512)(375,509)
(376,510)(377,497)(378,498)(379,500)(380,499)(381,503)(382,504)(383,501)
(384,502);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope