Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#538 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8)(9,15)(10,14)(11,13)(12,16)(17,19)(18,21)(20,22);(4,5,6)(7,9,11,17)(8,12,13,20)(10,16,18,22)(14,15,21,19);(8,13)(9,16)(12,19)(14,21)(15,20)(17,22);(8,14)(12,15)(13,21)(19,20)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8221);