Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #120 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#385 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ G | |
378 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)(10,11,15,19,16,20,14,27)(12,21,23,17,24,32,28,31)(13,18,25,34,26,35,29,33)(22,37,40,41,39,36,38,30);(10,12,15,23,16,24,14,28)(11,17,19,32,20,31,27,21)(13,22,25,40,26,39,29,38)(18,30,34,37,35,41,33,36);(10,13)(11,18)(12,22)(14,29)(15,25)(16,26)(17,30)(19,34)(20,35)(21,36)(23,40)(24,39)(27,33)(28,38)(31,41)(32,37)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 590);