Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #496225 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C2×#127 | C24 |
| prime radical | #6437452 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ C2×#3498 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,16)(5,11)(6,9)(7,15)(8,19)(10,22)(13,30)(14,28)(17,29)(18,27)(20,26)(21,24)(23,31)(25,32);(1,3,6,14)(2,7,9,24)(4,17,20,32)(5,21,22,15)(8,25,27,29)(10,28,11,12)(13,18,31,19)(16,30,26,23);(1,4)(2,8,11,26,9,27,10,16)(3,13,14,31)(5,18)(6,20)(7,23,12,29,24,30,28,25)(15,32,21,17)(19,22);(2,9)(3,14)(10,11)(13,31)(15,21)(17,32)(23,30)(25,29)Polycyclic presentation:
2^10|b^2=j,c^2=f,e^2=j,f^2=ij,h^2=j,b^a=be,c^a=cg,c^b=cei,d^a=deh,d^b=dj,d^c=dij,e^a=ej,e^b=ej,e^c=eij,f^a=fh,f^b=fi,g^b=gij,g^c=ghj,g^e=gj,g^f=gij,h^a=hj,h^b=hj,h^c=hi,h^g=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72317785, #72317880, #72318360, … |
| maximal normal subgroup of | #72317785, #72317880, #72318360, … |
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.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, 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, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*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.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);