Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1915 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #386 |
| commutator | C23 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#386 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
462 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)(10,11,14,19,15,18,20,33)(12,16,23,30,22,31,27,34)(13,24,26,35,25,17,28,32)(21,29,37,40,36,39,38,41);(10,12,20,23,15,22,14,27)(11,16,19,34,18,31,33,30)(13,21,26,38,25,36,28,37)(17,29,35,40,24,39,32,41);(10,13)(11,17)(12,21)(14,28)(15,25)(16,29)(18,24)(19,35)(20,26)(22,36)(23,38)(27,37)(30,41)(31,39)(32,33)(34,40)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 647);