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 | #485001 |
| commutator, Frattini, cosocle | C2×M32 | C24 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #33418 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
995 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)(5,24)(6,28)(7,9)(8,15)(10,22)(11,21)(12,26)(14,41)(16,40)(17,46)(19,38)(20,47)(23,57)(25,36)(27,58)(29,39)(30,43)(31,42)(32,52)(33,44)(34,54)(35,53)(37,63)(45,64)(48,56)(49,55)(50,61)(51,60)(59,62);(1,3)(2,8)(4,19)(5,25)(6,29)(7,16)(9,30)(10,36)(11,38)(12,31)(13,18)(14,26)(15,22)(17,21)(20,50)(23,51)(24,32)(27,59)(28,33)(34,62)(35,61)(37,64)(39,57)(40,47)(41,49)(42,63)(43,54)(44,56)(45,53)(46,48)(52,55)(58,60);(2,9)(8,30)(10,34)(11,35)(12,28)(15,43)(18,47)(19,50)(24,58)(31,33)(32,60)(37,48)(38,61)(42,44)(49,57)(51,52);(2,9)(3,14,16,29)(4,20)(8,31,30,33)(11,35)(12,28)(13,39,40,41)(15,44,43,42)(17,25,45,59)(19,51,50,52)(22,54)(23,55)(24,58)(32,61,60,38)(36,46,62,64)(56,63)Polycyclic presentation:
2^10|d^2=g,f^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=cgj,c^b=ce,d^a=dj,d^b=dij,e^a=eg,f^a=fh,f^b=fh,f^c=fg,f^d=fj,f^e=fj,g^b=gj,h^a=hi,h^b=hi,i^a=ij,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72342980 |
| maximal normal subgroup of | #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, 4, F.7);;
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, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
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);