Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2982 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C24 |
| commutator | C2 | C23×C8×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C23×C3×C4 |
| prime radical | C23×C3 | C22×C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,14,13,15,16,17)(18,19)(20,26)(21,24)(22,25)(23,28)(27,32)(29,31)(30,33);(18,20,22,28)(19,23,25,26)(21,29,30,27)(24,32,33,31);(18,21,22,30)(19,24,25,33)(20,27,28,29)(23,31,26,32);(19,25)(23,26)(24,33)(31,32)612 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1714);