Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#1371 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | D8 |
| commutator | C8 | C2×C32×C8 |
| Frattini, cosocle | C4×C8 | C22×C32 |
| socle, prime radical | C22×C32 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3)(7,8,9,11,10,12,13,14)(16,20)(17,21)(18,24)(22,30)(23,28)(25,29)(26,27);(4,5,6)(15,16,21,27,18,22,25,28,19,23,29,30,24,26,17,20)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2431);