Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#336 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C22 |
| commutator | C2 | C32×C4×C8 |
| Frattini, cosocle | C22×C4 | C22×C32 |
| socle, prime radical | C23×C32 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8)(9,10)(15,16,17,19,18,20,21,22);(4,5,6)(8,10)(11,12,13,14)360 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1331);