Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C23 |
| commutator | C22 | C24×C9 |
| Frattini, cosocle | C22×C3 | C24×C3 |
| socle, prime radical | C23×C3 | C23×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(15,16,17,18,19,20,21,22,23);(4,6)(7,8,11,14)(9,13,10,12);(7,9,11,10)(8,12,14,13);(3,4,5,6)(7,10,11,9)(8,13,14,12)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1845);