Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #486654 |
| commutator, Frattini, cosocle | C2×M32 | C24 |
| prime radical | #7624843 | C2 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33433 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
940 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)(5,9)(6,26)(7,28)(8,14)(10,22)(11,21)(12,25)(15,47)(16,49)(17,40)(19,53)(20,50)(23,39)(24,60)(27,41)(29,42)(30,44)(31,59)(32,62)(33,56)(34,55)(35,61)(36,43)(37,45)(38,46)(48,64)(51,58)(52,57)(54,63);(1,3,7,29,5,17,25,16)(2,8,12,38,9,30,28,32)(4,19,23,59,20,45,57,54)(6,27,21,15,24,61,55,48)(10,35,36,47,33,41,51,64)(11,37,26,31,34,53,60,63)(13,39,49,18,40,52,42,50)(14,43,62,22,44,58,46,56);(4,20)(8,30)(10,33)(13,40)(18,50)(22,56)(23,57)(31,63)(32,38)(35,41)(36,51)(37,53)(39,52)(42,49)(43,58)(47,64);(1,2)(3,8)(4,10)(5,9)(6,11)(7,12)(13,14)(15,31)(16,32)(17,30)(18,22)(19,35)(20,33)(21,26)(23,36)(24,34)(25,28)(27,37)(29,38)(39,43)(40,44)(41,45)(42,46)(47,59)(48,63)(49,62)(50,56)(51,57)(52,58)(53,61)(54,64)(55,60)Polycyclic presentation:
2^10|b^2=gij,f^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=cgj,c^b=ce,d^c=dgj,e^a=eg,e^d=ej,f^a=fh,f^b=fhj,f^c=fg,f^e=fj,g^b=gj,h^a=hi,h^b=hi,i^a=ij,i^b=ij67 conjugacy classes — view the full character table.
| maximal subgroup of | #72342960, #72342966, #72342980 |
| maximal normal subgroup of | #72342960, #72342966, #72342980 |
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, 2, F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);