Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C9×#535 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C25×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C23×C3 | C24×C3 |
| prime radical | C2×C3×#535 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
2844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(15,16,17,18,19,20,21,22,23);(4,6)(8,11)(9,13);(7,8,10,11)(9,12,13,14);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14);(7,9)(8,12)(10,13)(11,14)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2178);