Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C28 |
| direct product | C2×C9×#554 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C25×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C23×C3 | C24×C3 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4,7,6)(5,9,10,8)(11,12,15,14)(13,17,18,16);(4,6)(8,9)(11,13,15,18)(12,16,14,17);(3,5)(4,8)(6,9)(7,10);(3,6,7,4)(5,9,10,8)(11,14,15,12)(13,16,18,17)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2204);