Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×#120 |
| commutator, Frattini, cosocle | C22×C4 | C26 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
47975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(5,8)(10,14)(15,22)(16,20)(17,23);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15,14,22)(12,19)(13,18)(16,17,20,23)(21,24);(11,18)(15,22)(17,23)(19,24);(2,6)(5,8);(3,7)(5,8)(10,14)(12,21)(17,23)(19,24)Polycyclic presentation:
2^10|c^2=i,g^2=j,b^a=bj,c^a=ch,c^b=cg,d^c=dj,e^b=eh,f^b=fj,f^c=fhi,g^b=gj,g^f=gj,i^b=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72236707, #72236750, #72236784, … |
| maximal normal subgroup of | #72236707, #72236750, #72236784, … |
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, 3, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);