Finite Group DatabasebetaFinite Group Databasebeta| order | 32768 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C214 · Schur multiplier (H2) = C290 |
| extraspecial | 2−1+14 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C214 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
4073779669480638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 512 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 14 generators on 512 points:
(1,2,10,18)(3,19,26,11)(4,12,27,54)(5,13,33,60)(6,14,38,65)(7,15,42,69)(8,16,45,72)(9,17,47,74)(20,75,81,48)(21,76,87,49)(22,77,92,50)(23,78,96,51)(24,79,99,52)(25,80,101,53)(28,55,102,152)(29,56,103,153)(30,57,104,154)(31,58,105,155)(32,59,106,156)(34,61,117,167)(35,62,118,168)(36,63,119,169)(37,64,120,170)(39,66,127,177)(40,67,128,178)(41,68,129,179)(43,70,133,183)(44,71,134,184)(46,73,136,186)(82,187,202,137)(83,188,203,138)(84,189,204,139)(85,190,205,140)(86,191,206,141)(88,192,217,142)(89,193,218,143)(90,194,219,144)(91,195,220,145)(93,196,227,146)(94,197,228,147)(95,198,229,148)(97,199,233,149)(98,200,234,150)(100,201,236,151)(107,157,237,292)(108,158,238,293)(109,159,239,294)(110,160,240,295)(111,161,241,296)(112,162,242,297)(113,163,243,298)(114,164,244,299)(115,165,245,300)(116,166,246,301)(121,171,257,312)(122,172,258,313)(123,173,259,314)(124,174,260,315)(125,175,261,316)(126,176,262,317)(130,180,267,322)(131,181,268,323)(132,182,269,324)(135,185,271,326)(207,327,347,272)(208,328,348,273)(209,329,349,274)(210,330,350,275)(211,331,351,276)(212,332,352,277)(213,333,353,278)(214,334,354,279)(215,335,355,280)(216,336,356,281)(221,337,367,282)(222,338,368,283)(223,339,369,284)(224,340,370,285)(225,341,371,286)(226,342,372,287)(230,343,377,288)(231,344,378,289)(232,345,379,290)(235,346,381,291)(247,302,382,418)(248,303,383,419)(249,304,384,420)(250,305,385,421)(251,306,386,422)(252,307,387,423)(253,308,388,424)(254,309,389,425)(255,310,390,426)(256,311,391,427)(263,318,397,433)(264,319,398,434)(265,320,399,435)(266,321,400,436)(270,325,402,438)(357,439,454,403)(358,440,455,404)(359,441,456,405)(360,442,457,406)(361,443,458,407)(362,444,459,408)(363,445,460,409)(364,446,461,410)(365,447,462,411)(366,448,463,412)(373,449,469,413)(374,450,470,414)(375,451,471,415)(376,452,472,416)(380,453,474,417)(392,428,475,488)(393,429,476,489)(394,430,477,490)(395,431,478,491)(396,432,479,492)(401,437,481,494)(464,495,501,482)(465,496,502,483)(466,497,503,484)(467,498,504,485)(468,499,505,486)(473,500,507,487)(480,493,508,510)(506,511,512,509);(1,3,10,26)(2,11,18,19)(4,20,27,81)(5,21,33,87)(6,22,38,92)(7,23,42,96)(8,24,45,99)(9,25,47,101)(12,48,54,75)(13,49,60,76)(14,50,65,77)(15,51,69,78)(16,52,72,79)(17,53,74,80)(28,82,102,202)(29,83,103,203)(30,84,104,204)(31,85,105,205)(32,86,106,206)(34,88,117,217)(35,89,118,218)(36,90,119,219)(37,91,120,220)(39,93,127,227)(40,94,128,228)(41,95,129,229)(43,97,133,233)(44,98,134,234)(46,100,136,236)(55,137,152,187)(56,138,153,188)(57,139,154,189)(58,140,155,190)(59,141,156,191)(61,142,167,192)(62,143,168,193)(63,144,169,194)(64,145,170,195)(66,146,177,196)(67,147,178,197)(68,148,179,198)(70,149,183,199)(71,150,184,200)(73,151,186,201)(107,207,237,347)(108,208,238,348)(109,209,239,349)(110,210,240,350)(111,211,241,351)(112,212,242,352)(113,213,243,353)(114,214,244,354)(115,215,245,355)(116,216,246,356)(121,221,257,367)(122,222,258,368)(123,223,259,369)(124,224,260,370)(125,225,261,371)(126,226,262,372)(130,230,267,377)(131,231,268,378)(132,232,269,379)(135,235,271,381)(157,272,292,327)(158,273,293,328)(159,274,294,329)(160,275,295,330)(161,276,296,331)(162,277,297,332)(163,278,298,333)(164,279,299,334)(165,280,300,335)(166,281,301,336)(171,282,312,337)(172,283,313,338)(173,284,314,339)(174,285,315,340)(175,286,316,341)(176,287,317,342)(180,288,322,343)(181,289,323,344)(182,290,324,345)(185,291,326,346)(247,357,382,454)(248,358,383,455)(249,359,384,456)(250,360,385,457)(251,361,386,458)(252,362,387,459)(253,363,388,460)(254,364,389,461)(255,365,390,462)(256,366,391,463)(263,373,397,469)(264,374,398,470)(265,375,399,471)(266,376,400,472)(270,380,402,474)(302,403,418,439)(303,404,419,440)(304,405,420,441)(305,406,421,442)(306,407,422,443)(307,408,423,444)(308,409,424,445)(309,410,425,446)(310,411,426,447)(311,412,427,448)(318,413,433,449)(319,414,434,450)(320,415,435,451)(321,416,436,452)(325,417,438,453)(392,464,475,501)(393,465,476,502)(394,466,477,503)(395,467,478,504)(396,468,479,505)(401,473,481,507)(428,482,488,495)(429,483,489,496)(430,484,490,497)(431,485,491,498)(432,486,492,499)(437,487,494,500)(480,506,508,512)(493,509,510,511);(4,27)(12,54)(20,81)(28,102)(29,103)(30,104)(31,105)(32,106)(48,75)(55,152)(56,153)(57,154)(58,155)(59,156)(82,202)(83,203)(84,204)(85,205)(86,206)(107,237)(108,238)(109,239)(110,240)(111,241)(112,242)(113,243)(114,244)(115,245)(116,246)(137,187)(138,188)(139,189)(140,190)(141,191)(157,292)(158,293)(159,294)(160,295)(161,296)(162,297)(163,298)(164,299)(165,300)(166,301)(207,347)(208,348)(209,349)(210,350)(211,351)(212,352)(213,353)(214,354)(215,355)(216,356)(247,382)(248,383)(249,384)(250,385)(251,386)(252,387)(253,388)(254,389)(255,390)(256,391)(272,327)(273,328)(274,329)(275,330)(276,331)(277,332)(278,333)(279,334)(280,335)(281,336)(302,418)(303,419)(304,420)(305,421)(306,422)(307,423)(308,424)(309,425)(310,426)(311,427)(357,454)(358,455)(359,456)(360,457)(361,458)(362,459)(363,460)(364,461)(365,462)(366,463)(392,475)(393,476)(394,477)(395,478)(396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presentation:
2^15|a^2=o,b^2=o,b^a=bo,d^c=do,f^e=fo,h^g=ho,j^i=jo,l^k=lo,n^m=no16385 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n", "o");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.15);;
gap> SetPower(coll, 2, F.15);;
gap> SetConjugate(coll, 2, 1, F.2*F.15);;
gap> SetConjugate(coll, 4, 3, F.4*F.15);;
gap> SetConjugate(coll, 6, 5, F.6*F.15);;
gap> SetConjugate(coll, 8, 7, F.8*F.15);;
gap> SetConjugate(coll, 10, 9, F.10*F.15);;
gap> SetConjugate(coll, 12, 11, F.12*F.15);;
gap> SetConjugate(coll, 14, 13, F.14*F.15);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);