Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#2159 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| prime radical | C23×C32 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(7,8,9,10)(12,16)(13,17)(14,20)(18,22)(19,26)(21,25)(23,24);(4,5,6)(11,12,17,24,14,19,21,22,15,18,25,26,20,23,13,16);(12,18)(16,22)(19,23)(24,26)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3219);