Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#1359 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #331 |
| commutator | C42 | C2×C4×C9 |
| Frattini, cosocle | C3×C4×C8 | C22×C3 |
| socle | C2×C3 | C3×#331 |
| prime radical | C3×#127 | C3×C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C4×C8×C9 ◁ G | |
153 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,17,20,15,19,23,14,18,22,25,13,16,21,24);(11,15)(12,19)(13,20)(16,18)(17,25)(21,22)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 91);