Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C22×C4 |
| direct product | C9×#1978 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #121 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C3×#121 |
| prime radical | C22×C3×D8 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C22×C4×C9 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15,19,23,20,22,18,16)(17,25,27,29,28,24,26,21);(10,11,12,13)(14,16,19,22,20,23,18,15)(17,24,27,29,28,25,26,21);(14,17,19,27,20,28,18,26)(15,21,16,24,22,29,23,25)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 710);