Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#333 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | C22⋊C4 |
| commutator | C2×C4 | C2×C32×C4 |
| Frattini, cosocle | C42 | C22×C32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8,9,12,10,11,13,14)(16,19)(17,21)(20,22);(4,5,6)(8,11)(9,13)(15,16,18,20)(17,22,21,19)171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1328);