Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C28×C3 |
| direct product | C32×#3522 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
3312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2,3)(8,10)(12,13)(16,19)(20,22);(4,5,6)(7,8,11,10)(9,13,14,12);(15,16)(17,20)(18,19)(21,22);(7,9,11,14)(8,12,10,13);(7,10,11,8)(9,13,14,12)(15,17,18,21)(16,20,19,22)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4582);