Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C29 |
| direct product | C22×C9×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | C22 |
| commutator | C2 | C25×C9 |
| Frattini, cosocle | C2×C3 | C25×C3 |
| socle | C23×C3 | C23×C3 |
| prime radical | C22×C3×#41 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ G | |
1335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(13,14,15,16,17,18,19,20,21);(3,4);(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5031);