Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23×C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ G | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,5)(4,6)(7,8)(9,10,11);(2,5)(3,8)(6,7)(12,13,14);(2,6)(5,7)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 833);