Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#3275 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C3×#3275 | C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C4×D4×C9 ◁ G | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,18,21)(16,22,23,29)(17,26,27,20)(19,24,28,25);(14,16,18,23)(15,19,21,28)(17,24,27,25)(20,29,26,22);(11,13)(16,23)(19,28)(22,29)(24,25);(16,24)(17,27)(19,29)(20,26)(22,28)(23,25)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2007);