Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C23×C4 |
| direct product | C9×#2499 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C23 | C24×C9 |
| Frattini, socle, cosocle | C23×C3 | C24×C3 |
| prime radical | C24×C3 | C23×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
726 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)(11,13)(15,18)(16,20)(22,23,26,28)(24,25,27,29);(14,15)(16,19)(17,18)(20,21)(22,24)(23,25)(26,27)(28,29);(10,11,12,13)(22,25)(23,27)(24,28)(26,29);(14,16,17,20)(15,19,18,21)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1231);