Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C24 |
| commutator | C2 | C23×C32×C4 |
| Frattini, cosocle | C4 | C24×C32 |
| socle | C2×C32 | C23×C4 |
| prime radical | C2×C32×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8,10,14,11,13,17,20)(9,12,16,19,15,18,21,22);(4,5,6)(7,9)(8,12)(10,16)(11,15)(13,18)(14,19)(17,21)(20,22);(9,15)(12,18)(16,21)(19,22);(8,13)(12,18)(14,20)(19,22)306 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8251);