Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C9 · Schur multiplier (H2) = C4 |
| direct product | C9×#322 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C9 | C22 |
| commutator | C2 | C4×C8×C9 |
| Frattini, cosocle | C3×C42 | C22×C3 |
| socle, prime radical | C22×C3 | C3×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C9 ◁ G | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,15,13,14,16,17);(11,14)(15,17)(18,19,20,22,21,23,24,25)360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 128);