Polytope of Type {8,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*512s
if this polytope has a name.
Group : SmallGroup(512,58358)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 32, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
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}*256d, {4,8}*256d, {8,8}*256e
   4-fold quotients : {8,4}*128a, {4,8}*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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 2.
      16 facets:
         16 of {8}*16
      24 vertex figures:
         8 of {8}*16
         16 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      8 facets:
         8 of {8}*16
      16 vertex figures:
         4 of {8}*16
         4 of {4}*8
         8 of {2}*4

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

Twisty Puzzle