Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3 |
| direct product | C22×C32×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C32 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| prime radical | C25×C32 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C32 ◁ G | |
5520 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)(17,18,19);(3,4)(20,21,22);(6,8)(9,10,12,14)(11,15,16,13);(9,11,12,16)(10,13,14,15);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4493);