Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C22×C4 |
| direct product | C9×#1975 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C3×C4×D4 |
| prime radical | C2×C3×#135 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C22×C8×C9 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15,18,22,19,23,17,20)(16,24,27,29,25,28,26,21);(10,11,12,13)(15,20,23,22)(16,25)(21,24,29,28)(26,27);(14,16,17,26,19,25,18,27)(15,21,22,24,23,29,20,28)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 707);