Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2931 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C9 | C23 |
| commutator | C22 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C23×C3×C4 |
| prime radical | C3×#532 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C9 ◁ G | |
750 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)(14,15,19,22,18,21,27,29)(16,23,25,26,24,20,28,17);(11,13)(14,16)(15,20)(17,22)(18,24)(19,25)(21,23)(26,29)(27,28);(14,17,19,23,18,26,27,20)(15,16,22,25,21,24,29,28);(15,21)(16,24)(22,29)(25,28)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1663);