Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#362 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #102 |
| commutator | C8 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C3 |
| socle | C2×C3 | C3×#102 |
| prime radical | C22×C3 | C3×SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C2×C8×C9 ◁ G | |
81 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)(10,11,14,18,15,20,30,19)(12,21,23,34,24,38,29,37)(13,25,27,17,28,35,41,36)(16,26,31,22,33,39,32,40);(10,12,15,24)(11,16,20,33)(13,26,28,39)(14,29,30,23)(17,34,36,37)(18,32,19,31)(21,25,38,35)(22,27,40,41)144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 168);