Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C26×C3 |
| direct product | C32×#1439 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C32 | C23 |
| commutator | C23 | C22×C32×C4 |
| Frattini, cosocle | C24 | C23×C32 |
| prime radical | C25×C32 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C32 ◁ G | |
2340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(12,14)(15,16,17,18)(20,22);(4,5,6)(8,10)(12,14)(19,20,21,22);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2499);