Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#551 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C24 |
| commutator | C22 | C24×C9 |
| Frattini, socle, cosocle | C22×C3 | C24×C3 |
| prime radical | C3×#120 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,17)(12,18,19,15)(14,16,21,23)(20,24,25,22);(10,12,14,20)(11,15,16,22)(13,19,21,25)(17,18,23,24);(11,16)(15,22)(17,23)(18,24);(12,19)(15,18)(20,25)(22,24)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1865);