Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C16 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×Q8⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C8 | D4 |
| commutator | C4 | C2×C32×C16 |
| Frattini, cosocle | C4×C8 | C22×C32 |
| socle, prime radical | C22×C32 | C22⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C8 ◁ C32×C4×C8 ◁ G | |
282 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3)(8,11)(9,13)(12,14)(15,16,17,20,18,21,23,26,19,22,24,27,25,28,29,30);(4,5,6)(7,8,10,12)(9,14,13,11)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2397);