Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#2253 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×D8 |
| commutator | C2×C8 | C23×C9 |
| Frattini, cosocle | C2×C3×C8 | C23×C3 |
| socle, prime radical | C22×C3 | C2×C3×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C42×C9 ◁ C4×C8×C9 ◁ G | |
174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,15)(12,16,17,14)(18,19,21,24)(20,28,30,42)(22,35,33,48)(23,36,34,27)(25,39,37,31)(26,41,38,29)(32,47,46,44)(40,45,43,49);(10,12,13,17)(11,14,15,16)(19,24)(25,37)(26,38)(27,36)(28,42)(35,48)(40,43)(44,47);(18,20,29,45,23,32,33,39,21,30,41,49,34,46,22,31)(19,25,35,44,27,40,38,42,24,37,48,47,36,43,26,28)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 985);