Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C9 · Schur multiplier (H2) = C24 |
| direct product | C2×C9×#323 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C22⋊C4 |
| commutator | C22 | C22×C8×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C23×C3 | C3×C22⋊C4 |
| prime radical | C25×C3 | C3×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C23×C4×C9 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4,5,9,7,10,12,16)(6,8,11,14,13,15,17,18);(4,8)(5,11)(10,15)(12,17)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 188);