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×#358 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | D8 |
| commutator | C8 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C3 |
| socle, prime radical | C22×C3 | C3×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C2×C8×C9 ◁ G | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13)(15,19)(16,22)(17,24)(20,29)(21,28)(23,26)(25,27);(14,15,18,21)(16,23,25,29)(17,26,24,20)(19,27,28,22)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 164);