Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C9×#124 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C9 | C23 |
| commutator | C22 | C25×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C24×C3 | C23×C3 |
| prime radical | C23×C3×D4 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C9 ◁ G | |
2820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(17,18,19,20,21,22,23,24,25);(3,4);(6,7)(11,15)(13,16);(5,6,8,7)(9,10,12,14)(11,13,15,16);(5,7,8,6)(9,11)(10,13)(12,15)(14,16)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2168);