Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | D32 |
| commutator | C32 | C22×C9 |
| Frattini, cosocle | C3×C32 | C22×C3 |
| socle | C2×C3 | C3×D32 |
| prime radical | C3×D32 | C2×C3 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C4×C9 ◁ C8×C9 ◁ C9×C16 ◁ C9×C32 ◁ G | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(11,41)(12,72)(13,39)(14,70)(15,37)(16,68)(17,35)(18,66)(19,33)(20,64)(21,31)(22,62)(23,29)(24,60)(25,27)(26,58)(28,56)(30,54)(32,52)(34,50)(36,48)(38,46)(40,44)(43,73)(45,71)(47,69)(49,67)(51,65)(53,63)(55,61)(57,59)315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 162);