Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#1335 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | D8 |
| commutator | C8 | C2×C8×C9 |
| Frattini, cosocle | C3×C4×C8 | C22×C3 |
| socle | C2×C3 | C3×D4⋊C8 |
| prime radical | C3×#135 | C3×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ C2×C8×C9 ◁ C4×C8×C9 ◁ G | |
150 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 66 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)(10,11,14,19,16,20,25,18,17,21,13,24,15,12,22,23);(10,12)(11,16)(13,18)(14,23)(15,21)(17,20)(19,25)(22,24)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 67);