Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C29 |
| direct product | C2×C9×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C24 |
| commutator | C2 | C25×C9 |
| Frattini, cosocle | C2×C3 | C25×C3 |
| socle | C22×C3 | C24×C3 |
| prime radical | C2×C3×2−1+4 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
1191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(4,8)(6,14)(9,11)(12,18);(3,4,7,8)(5,11,13,9)(6,10,14,15)(12,17,18,16);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17);(3,6)(4,10)(5,12)(7,14)(8,15)(9,16)(11,17)(13,18)306 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5033);