Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#401 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C23 |
| commutator, Frattini, cosocle | C23 | C23×C32 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(9,13)(11,14)(16,19)(20,21)(23,24,26,28)(25,29,30,27);(4,5,6)(7,8,10,12)(9,14,13,11)(15,16,18,19)(17,21,22,20)(23,24)(25,29)(26,28)(27,30);(7,9)(8,11)(10,13)(12,14)(15,17,18,22)(16,20,19,21)(23,25,26,30)(24,27,28,29)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4856);