Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1277 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | D4 |
| commutator | C4 | C4×C8×C9 |
| Frattini, cosocle | C2×C3×C42 | C22×C3 |
| socle, prime radical | C23×C3 | C3×C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ C2×C42×C9 ◁ G | |
270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(15,19)(16,17)(20,21)(22,23,24,26,25,27,28,29);(11,13)(14,15,17,20,18,21,16,19)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 9);