Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23×D4 |
| commutator | C22×C4 | C24×C4 |
| Frattini, cosocle | C23×C4 | C25 |
| socle | C23 | C22×C4×D4 |
| prime radical | #6442878 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C8×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,12,15,13,14,20,17)(11,18,21,22,19,23,24,16);(2,4)(5,6)(7,8)(10,14)(15,17)(16,22)(18,23);(6,8);(10,15,14,17)(11,19)(16,23,22,18)(21,24);(9,11,20,24,13,19,12,21)(10,16,15,18,14,22,17,23)Polycyclic presentation:
2^10|a^2=f,d^2=fgi,e^2=ij,f^2=j,i^2=j,b^a=bg,c^b=ch,d^a=di,e^a=eij,e^d=ej,i^a=ij172 conjugacy classes — view the full character table.
| maximal subgroup of | #72309712, D4×#21546, #72317307, … |
| maximal normal subgroup of | #72309712, D4×#21546, #72317307, … |
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.6*F.7*F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);