Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#554 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| prime radical | C2×C32×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8,11,10)(9,13,14,12)(15,16,19,18)(17,21,22,20);(4,5,6)(8,10)(12,13)(15,17,19,22)(16,20,18,21);(7,9)(8,12)(10,13)(11,14);(7,10,11,8)(9,13,14,12)(15,18,19,16)(17,20,22,21)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8237);