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 | C9×#3480 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C24 |
| commutator | C22 | C25×C9 |
| Frattini, socle, cosocle | C22×C3 | C25×C3 |
| prime radical | C3×#3480 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(11,14)(15,17)(18,19,22,24)(20,23,21,25);(10,11)(12,15)(13,14)(16,17)(18,20,22,21)(19,23,24,25);(18,21,22,20)(19,23,24,25);(11,14)(15,17);(10,12,13,16)(11,15,14,17)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2212);