Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2913 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C9×#41 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13)(15,19)(17,20)(21,25)(22,26)(24,27)(28,29);(14,15,20,26,18,22,17,19)(16,21,27,29,23,28,24,25);(14,16)(15,21)(17,24)(18,23)(19,25)(20,27)(22,28)(26,29);(16,23)(21,28)(24,27)(25,29)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1645);