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×#13082 |
| commutator | C22×Q8 | C23×C4 |
| Frattini, cosocle | C2×#124 | C24 |
| prime radical | #6437458 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3957 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)(3,13)(4,18,22,55,20,54,24,36)(5,12)(6,26,21,38)(7,11)(8,14)(9,23,34,58,33,57,35,27)(10,28,37,29)(15,45,46,63,19,53,48,64)(16,49,43,59)(17,42)(25,39)(30,56,61,50,40,47,62,60)(31,52,41,51)(32,44);(1,3)(2,8)(4,19)(5,14)(6,16)(7,17)(9,30)(10,31)(11,32)(12,13)(15,20)(18,53)(21,43)(22,48)(23,56)(24,46)(25,44)(26,59)(27,60)(28,51)(29,52)(33,40)(34,61)(35,62)(36,63)(37,41)(38,49)(39,42)(45,54)(47,57)(50,58)(55,64);(3,14)(4,20)(8,13)(9,33)(16,43)(17,44)(18,54)(22,24)(23,57)(27,58)(31,41)(32,42)(34,35)(36,55)(49,59)(51,52);(1,4,21,58)(2,9,10,36)(3,15,43,60)(5,20,6,27)(7,22,28,57)(8,30,31,63)(11,34,38,18)(12,33,37,55)(13,40,41,64)(14,19,16,50)(17,46,51,56)(23,25,24,29)(26,54,39,35)(32,61,49,53)(42,62,59,45)(44,48,52,47)Polycyclic presentation:
2^10|a^2=e,d^2=efgi,e^2=i,g^2=j,h^2=j,b^a=bij,c^b=cj,d^a=dg,d^b=df,d^c=dj,e^d=eh,f^a=fj,g^a=gh,g^d=ghj,g^e=gij,h^a=hi,h^d=hi,h^g=hj,i^d=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72306296, #72306356, #72306381, … |
| maximal normal subgroup of | #72306296, #72306356, #72306381, … |
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.5);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);