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×C4 |
| direct product | C9×#353 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #99 |
| commutator | C2×C4 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×D4 | C22×C3 |
| socle | C2×C3 | C3×#99 |
| prime radical | C3×#127 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C9 ◁ C2×D4×C9 ◁ G | |
129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,15,17,14)(13,16);(11,14)(12,17)(15,16)117 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 159);