Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×C9×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C2×C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ C2×C42×C9 ◁ G | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(23,24,25,26,27,28,29,30,31);(3,4,5,6);(7,8,9,10,11,12,13,14)(15,22,21,20,19,18,17,16);(7,15,11,19)(8,16,12,20)(9,17,13,21)(10,18,14,22)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1670);