Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1278 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C42 | D4 |
| commutator | C4 | C32×C4×C8 |
| Frattini, cosocle | C2×C42 | C22×C32 |
| socle | C22×C32 | #95 |
| prime radical | C23×C32 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C42 ◁ C2×C32×C42 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8)(9,11)(10,13)(12,14)(15,16,18,21,20,22,19,17);(4,5,6)(8,11,13,12)(15,17,19,22,20,21,18,16)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2338);