Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | D16 |
| commutator | C16 | C22×C9 |
| Frattini, cosocle | C3×C16 | C22×C3 |
| socle | C2×C3 | C3×D16 |
| prime radical | C3×D16 | C2×C3 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C4×C9 ◁ C8×C9 ◁ C9×C16 ◁ G | |
48 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)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(11,25)(12,40)(13,23)(14,38)(15,21)(16,36)(17,19)(18,34)(20,32)(22,30)(24,28)(27,41)(29,39)(31,37)(33,35)171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 178);