Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C9 · Schur multiplier (H2) = C29 |
| direct product | C2×C9×#519 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C2 | C24×C4×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C23×C3 | C24×C3 |
| prime radical | C24×C3 | C23×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
2232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(23,24,25,26,27,28,29,30,31);(3,4,5,6)(8,12)(10,18)(13,15)(16,22);(7,8,11,12)(9,15,17,13)(10,14,18,19)(16,21,22,20);(7,9,11,17)(8,13,12,15)(10,16,18,22)(14,20,19,21);(7,10)(8,14)(9,16)(11,18)(12,19)(13,20)(15,21)(17,22)612 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2158);