Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C9 · Schur multiplier (H2) = C22×C4 |
| direct product | C9×#1984 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | C23 |
| commutator | C22 | C22×C8×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle, prime radical | C23×C3 | C22×C3×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15)(16,20)(17,18)(19,21)(22,23,25,28,26,24,27,29);(10,11,12,13)(15,18)(19,20)(22,24,25,29,26,23,27,28);(14,16,17,21)(15,19,18,20)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 716);