Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#430 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C2×D4 |
| commutator | C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C4 | C23×C32 |
| socle | C2×C32 | C2×C4⋊C4 |
| prime radical | C23×C32 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8,11,14,12,15,10,20)(9,16,19,21,17,22,18,13);(4,5,6)(7,9,10,18,12,17,11,19)(8,13,14,16,15,21,20,22);(9,17)(13,21)(16,22)(18,19)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4885);