Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C9 · Schur multiplier (H2) = C211 |
| direct product | C23×C9×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C9 | C22 |
| commutator | C2 | C24×C4×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C25×C3 | C22×C3 |
| prime radical | C26×C3 | C2×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C9 ◁ G | |
6948 conjugacy classes of subgroups; only the 2964 normal subgroups are shown.
This group has 6948 conjugacy classes of subgroups — too many to draw. Only its 2964 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(15,16,17,18,19,20,21,22,23);(3,4);(5,6);(7,8,10,12)(9,11,13,14);(8,11)(12,14)720 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2151);