Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#1333 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #102 |
| commutator | C8 | C2×C8×C9 |
| Frattini, cosocle | C3×C4×C8 | C22×C3 |
| socle | C22×C3 | C3×#102 |
| prime radical | C2×C3×D8 | C3×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C42×C9 ◁ C4×C8×C9 ◁ G | |
162 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,14,13,15,16,17)(18,19,21,25,22,27,32,26)(20,28,30,24,31,23,29,33);(19,23)(20,29)(21,32)(24,25)(26,33)(27,28)(30,31)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 65);