Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#547 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C24 |
| commutator | C22 | C24×C9 |
| Frattini, socle, cosocle | C22×C3 | C24×C3 |
| prime radical | C3×#547 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15,18,19)(16,17,20,21);(15,19)(17,21);(10,11)(12,13)(14,16,18,20)(15,17,19,21);(14,17)(15,20)(16,19)(18,21)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1861);