Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#1292 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #95 |
| commutator | C2×C4 | C42×C9 |
| Frattini, cosocle | C2×C3×C42 | C22×C3 |
| socle | C22×C3 | C3×#95 |
| prime radical | C23×C3 | C3×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C42×C9 ◁ G | |
228 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 576 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(11,15)(12,13)(16,17)(18,19,20,23,21,22,24,25);(10,11,13,16,14,17,12,15)(19,22)(20,24)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 24);