Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×#1426 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #367 |
| commutator | C16 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C16 | C22×C3 |
| socle | C2×C3 | C3×#367 |
| prime radical | C22×C3 | C3×SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C2×C8×C9 ◁ C2×C9×C16 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(11,16,18,29)(12,21)(13,25)(14,26)(17,32)(19,33,36,41)(20,39,37,31)(22,27)(23,35)(24,34)(28,40,38,30);(10,11,17,33,13,19,12,16,26,37,21,30,22,28,32,39,15,18,35,41,27,36,24,29,14,20,34,40,25,38,23,31)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 158);