Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C22×C3×AGL(1,ℤ/8) (order 384) · Inn(G) = D4 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | D4 |
| commutator | C4 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C3 |
| socle | C2×C3 | C3×C4⋊C4 |
| prime radical | C22×C3 | C3×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C2×C4×C9 ◁ G | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18,20,19,21,23,22,24,25);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 58);