Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#2498 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C23 | C24×C9 |
| Frattini, socle, cosocle | C23×C3 | C24×C3 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
819 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)(15,16)(17,21)(22,23,26,28)(24,25,29,27);(14,15,18,16)(17,19,21,20)(22,24,26,29)(23,27,28,25);(10,11,12,13)(14,16,18,15)(17,20,21,19)(22,24)(23,25)(26,29)(27,28);(14,17)(15,19)(16,20)(18,21)(22,25)(23,24)(26,27)(28,29)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1230);