Polytope of Type {16,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,8}*512d
if this polytope has a name.
Group : SmallGroup(512,32838)
Rank : 3
Schlafli Type : {16,8}
Number of vertices, edges, etc : 32, 128, 16
Order of s0s1s2 : 16
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}*256a
   4-fold quotients : {4,8}*128a, {8,4}*128a, {8,8}*128a, {8,8}*128b, {8,8}*128c, {8,8}*128d
   8-fold quotients : {4,8}*64a, {8,4}*64a, {4,8}*64b, {8,4}*64b, {4,4}*64
   16-fold quotients : {4,4}*32, {2,8}*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*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      8 facets:
         8 of {16}*32
      16 vertex figures:
         16 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 2.
      8 facets:
         8 of {16}*32
      16 vertex figures:
         16 of {8}*16

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

Twisty Puzzle