Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×#340 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C9 | D4 |
| commutator | C4 | C2×C42×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle, prime radical | C24×C3 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C9 ◁ C23×C4×C9 ◁ G | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4)(5,6)(7,8,9,10)(12,16)(13,14)(17,18);(4,6)(11,12,14,17,15,18,13,16)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 466);