Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#1379 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C9 | D4 |
| commutator | C4 | C4×C8×C9 |
| Frattini, cosocle | C22×C3×C8 | C22×C3 |
| socle | C22×C3 | C3×C22⋊C8 |
| prime radical | C23×C3 | C3×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C9 ◁ C22×C8×C9 ◁ G | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9)(14,15,18,23,19,24,33,41,20,25,34,42,17,22,31,39)(16,26,28,43,27,37,44,40,29,38,32,45,30,21,35,36);(10,11,12,13)(14,16)(15,21)(17,30)(18,32)(19,27)(20,29)(22,38)(23,40)(24,26)(25,37)(28,34)(31,44)(33,35)(36,42)(39,43)(41,45)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 111);