Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#546 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2,3)(8,10)(11,12,15,16)(13,17,18,14);(4,5,6)(12,16)(14,17);(7,8)(9,10)(11,13)(12,14)(15,18)(16,17);(11,14)(12,13)(15,17)(16,18)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8229);