Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#355 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #99 |
| commutator | C2×C4 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×Q8 | C22×C3 |
| socle | C2×C3 | C3×#99 |
| prime radical | C2×C3×D4 | C3×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C9 ◁ C2×Q8×C9 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,17,14,18,22,25)(12,16,20,24,19,15,21,23);(11,15)(12,19)(16,18)(17,24)(20,21)(23,25)117 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 161);