Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C9 · Schur multiplier (H2) = C23×C4 |
| direct product | C9×#1735 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | D4 |
| commutator | C4 | C2×C42×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C23×C3 | C2×C3×D4 |
| prime radical | C24×C3 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ C23×C4×C9 ◁ G | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,16,17)(18,19)(20,28)(21,25)(22,26)(23,27)(24,29)(30,33)(31,32);(11,13)(18,20,22,30,23,31,21,29)(19,24,25,32,27,33,26,28);(18,21,23,22)(19,25,27,26)(20,29,31,30)(24,32,33,28)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 467);