Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1909 |
| 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 | |
468 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)(12,20)(13,24,26,17)(14,27)(15,19)(16,22)(18,29)(21,34,37,41)(23,35)(25,32,28,33)(30,36,40,38)(31,39);(10,12,14,22,15,23,29,39)(11,16,18,20,19,31,27,35)(13,21,28,38,26,37,25,36)(17,30,33,34,24,40,32,41);(10,13)(11,17)(12,21)(14,28)(15,26)(16,30)(18,33)(19,24)(20,34)(22,38)(23,37)(25,29)(27,32)(31,40)(35,41)(36,39)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 641);