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) = C22×C3 |
| direct product | C32×#338 |
| 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 | Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
258 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)(7,8,10,14,12,15,11,20)(9,16,18,13,19,21,17,22);(4,5,6)(7,9,11,18,12,19,10,17)(8,13,14,16,15,22,20,21)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1333);