Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #102 |
| commutator | C8 | C2×C32×C4 |
| Frattini, cosocle | C2×C8 | C22×C32 |
| prime radical | C2×C32×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C2×C32×C8 ◁ G | |
186 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)(7,8,10,14,11,16,21,15)(9,17,19,13,20,12,18,22);(4,5,6)(8,12)(9,18)(10,21)(13,14)(15,22)(16,17)(19,20)144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1356);