Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1916 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | C23 |
| commutator | C22 | C22×C8×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle, prime radical | C23×C3 | C22×C3×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15)(16,19)(17,18)(20,21)(22,23,27,24,26,29,25,28);(15,18)(19,21)(22,24,27,29,26,28,25,23);(11,13)(14,16,17,20)(15,19,18,21)(22,25,26,27)(23,28,29,24)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 648);