Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1813 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #117 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#380 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C2×C9×M16 ◁ 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)(10,11,15,19,16,20,14,29)(12,21,24,32,25,36,23,17)(13,26,27,33,28,18,30,34)(22,37,38,41,39,35,40,31);(10,12,14,23,16,25,15,24)(11,17,19,21,20,32,29,36)(13,22,27,38,28,39,30,40)(18,31,33,35,26,41,34,37);(10,13)(11,18)(12,22)(14,27)(15,30)(16,28)(17,31)(19,33)(20,26)(21,35)(23,38)(24,40)(25,39)(29,34)(32,41)(36,37)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 545);