Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C9 · Schur multiplier (H2) = C28 |
| direct product | C22×C9×#95 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C9 | C22 |
| commutator | C2 | C22×C42×C9 |
| Frattini, cosocle | C23×C3 | C24×C3 |
| socle, prime radical | C25×C3 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C9 ◁ G | |
2820 conjugacy classes of subgroups; only the 1668 normal subgroups are shown.
This group has 2820 conjugacy classes of subgroups — too many to draw. Only its 1668 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(17,18,19,20,21,22,23,24,25);(3,4);(10,12)(13,14,16,15);(5,6,7,8)(9,10,11,12)(13,15,16,14)720 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 998);