Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1294 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C22⋊C4 |
| commutator | C2×C4 | C42×C9 |
| Frattini, cosocle | C23×C3×C4 | C22×C3 |
| socle | C23×C3 | C3×C22⋊C4 |
| prime radical | C24×C3 | C3×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C23×C4×C9 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13)(14,15)(16,18)(17,20)(19,21)(22,23)(24,28)(25,27)(26,29);(15,18,20,19)(22,24)(23,26,27,28)(25,29)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 26);