Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1902 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #386 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#386 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,21)(13,25,28,35)(14,29)(15,19)(16,20)(17,23)(18,27,34,26)(22,37,39,31)(24,32)(30,33)(36,38,41,40);(10,12,14,23,16,24,15,30)(11,17,19,21,20,33,29,32)(13,22,27,40,28,39,26,38)(18,31,35,36,34,37,25,41);(10,13)(11,18)(12,22)(14,27)(15,26)(16,28)(17,31)(19,35)(20,34)(21,36)(23,40)(24,39)(25,29)(30,38)(32,41)(33,37)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 634);