Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#359 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | D8 |
| commutator | C8 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C3 |
| socle | C2×C3 | C3×#102 |
| prime radical | C3×#135 | C2×C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C2×C4×C9 ◁ C2×C8×C9 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,19,16,20,15,30)(12,21,23,37,24,38,31,35)(13,25,27,36,29,40,28,18)(17,26,32,39,34,41,33,22);(10,12)(11,17)(13,26)(14,23)(15,31)(16,24)(18,35)(19,32)(20,34)(21,25)(22,28)(27,39)(29,41)(30,33)(36,37)(38,40)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 165);