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 | C2×C4 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C26 |
| socle | C22 | C23×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
40679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(12,20)(14,22)(15,21)(19,24);(9,10)(11,17)(12,21)(13,16)(14,19)(15,20)(18,23)(22,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14,16,22)(12,19)(13,18)(15,17,21,23)(20,24);(1,3,4,7)(2,5,6,8);(2,6)(5,8)(11,18)(14,22)(17,23)(19,24);(2,6)(5,8)Polycyclic presentation:
2^10|c^2=h,d^2=i,g^2=j,c^a=ch,c^b=cg,e^c=eij,f^c=fi,g^a=gj,g^b=gj,h^b=hj184 conjugacy classes — view the full character table.
| maximal subgroup of | #72239488, #72257590, #72278713, … |
| maximal normal subgroup of | #72239488, #72257590, #72278713, … |
| maximal quotient of | #72264580, #72264663 |
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.8);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);