Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×C32×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C32 |
| socle | C22×C32 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(6,7,8);(3,4,5)(10,13)(14,16);(9,10)(11,14)(12,13)(15,16);(1,2)(9,11,12,15)(10,14,13,16)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1021);