Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C23 |
| direct product | C2×C9×#352 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #99 |
| commutator | C23 | C22×C4×C9 |
| Frattini, cosocle | C2×C3×D4 | C23×C3 |
| socle | C22×C3 | C3×#99 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C23×C9 ◁ C22×D4×C9 ◁ G | |
417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4)(5,11)(6,14,15,9)(7,10)(8,12,16,18)(13,17);(3,5,7,13)(4,8,10,16)(6,12,15,18)(9,11,14,17)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 851);