Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C42 |
| commutator | C4 | C32×C42 |
| Frattini, cosocle | C22×C4 | C22×C32 |
| socle | C2×C32 | #95 |
| prime radical | C23×C32 | D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8)(9,15,17,21)(10,13)(11,14)(12,16,19,22)(18,20);(4,5,6)(7,9)(8,12,14,19)(10,16)(11,17)(13,15,20,21)(18,22)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1332);