Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C16 · Schur multiplier (H2) = C2 |
| direct product | C9×C4⋊C16 |
| unambiguous semidirect product | C4⋊(C9×C16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C9 | C22 |
| commutator | C2 | C2×C9×C16 |
| Frattini, cosocle | C2×C3×C8 | C22×C3 |
| socle, prime radical | C22×C3 | C2×C3×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C9 ◁ G | |
75 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)(11,13)(14,15,16,19,17,20,22,25,18,21,23,26,24,27,28,29);(10,11,12,13)360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 169);