Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13086 |
| commutator | C2×#41 | C23×C4 |
| Frattini, cosocle | #532 | C24 |
| prime radical | C2×#23873 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
4069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8,16,31,17,29,18,32)(4,19,21,47)(5,23)(6,26)(7,10)(9,20,33,50)(11,24)(12,13)(14,39,41,61,42,60,46,64)(15,30,43,58,45,52,44,59)(22,48,27,49)(25,36)(28,54,56,62,55,63,57,40)(34,53,37,51)(35,38);(1,3,13,16)(2,8)(4,14,22,41)(5,15,25,44)(6,18,7,17)(9,28)(10,29)(11,30,23,52)(12,32,26,31)(19,39,47,60)(20,40,53,63)(21,42,27,46)(24,45,38,43)(33,55)(34,56,37,57)(35,58)(36,59)(48,64)(49,61)(50,62,51,54);(1,4)(2,9)(3,14)(5,20)(6,27)(7,21)(8,28)(10,33)(11,19)(12,37)(13,22)(15,40)(16,41)(17,42)(18,46)(23,47)(24,50)(25,53)(26,34)(29,55)(30,39)(31,56)(32,57)(35,49)(36,48)(38,51)(43,54)(44,63)(45,62)(52,60)(58,61)(59,64);(2,10)(8,29)(9,33)(11,23)(12,26)(19,47)(28,55)(30,52)(31,32)(34,37)(35,36)(39,60)(48,49)(56,57)(58,59)(61,64)Polycyclic presentation:
2^10|a^2=f,b^2=ghi,f^2=ij,h^2=j,b^a=bfg,c^a=ce,d^a=dj,e^a=ei,e^b=ei,f^b=fh,f^c=fi,g^a=ghj,g^b=gij,g^c=gj,g^e=gj,g^f=gij,h^a=hi,h^b=hi,h^c=hj,h^g=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72286971, #72287007, #72287010, … |
| maximal normal subgroup of | #72286971, #72287007, #72287010, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);