Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C23×C4 |
| direct product | C9×#1899 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C9 |
| Frattini, cosocle | C24×C3 | C23×C3 |
| socle | C22×C3 | C2×C3×C22⋊C4 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C23×C4×C9 ◁ G | |
567 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)(10,11)(12,13)(14,15)(16,19,23,26)(17,20)(18,21)(22,25,28,29)(24,27);(14,16)(15,19,21,26)(17,22)(18,23)(20,25,27,29)(24,28);(11,13)(14,17,18,24)(15,20,21,27)(16,22,23,28)(19,25,26,29)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 631);