Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #33669 |
| commutator | C2×C16 | C23×C4 |
| Frattini, cosocle | C22×C16 | C24 |
| socle | C2 | #495688 |
| prime radical | #512840 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C4×C8 ◁ C2×C4×M32 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,4,11,20,36,54,61,6,10,22,35,25,34,52,41,7,9,19,33,51,60,64,63,24,32,53,38,5,12,21,39)(3,13,15,43,23,56,50,28,17,42,47,30,46,37,48,8,14,29,45,40,57,62,55,31,26,58,49,59,16,44,18,27);(1,3)(2,8)(4,18)(5,23)(6,26)(7,14)(9,27)(10,28)(11,37)(12,40)(13,41)(15,21)(16,20)(17,24)(19,48)(22,55)(25,57)(29,39)(30,36)(31,32)(33,44)(34,43)(35,56)(38,62)(42,61)(45,52)(46,51)(47,64)(49,54)(50,53)(58,63)(59,60);(2,9)(4,19)(8,27)(10,32)(12,34)(15,45)(18,48)(21,52)(22,53)(28,31)(30,59)(36,60)(40,43)(47,49)(50,55)(54,64);(2,10,9,32)(4,19)(8,28,27,31)(11,38,33,35)(12,36,34,60)(13,42,29,58)(15,45)(18,48)(21,52)(22,53)(30,43,59,40)(37,62,44,56)(39,63,41,61)(47,49)(50,55)(54,64)Polycyclic presentation:
2^10|a^2=f,d^2=g,f^2=hi,h^2=ij,i^2=j,b^a=bfj,c^a=ce,d^a=dij,e^a=ej,e^b=ej,f^b=fh,f^c=fj,f^d=fj,g^a=gj,h^b=hi,i^b=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72225019, #72225065, #72225073 |
| maximal normal subgroup of | #72225019, #72225065, #72225073 |
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, 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*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);