Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×#470 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C2×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C23×C3 | C2×C3×D4 |
| prime radical | C24×C3 | C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C23×C4×C9 ◁ G | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(23,24,25,26,27,28,29,30,31);(3,4)(5,6)(7,8,11,14)(9,15,17,22)(10,19,16,13)(12,20,21,18);(7,9,16,20,11,17,10,18)(8,12,19,22,14,21,13,15);(4,6)(9,17)(12,21)(15,22)(18,20)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1785);