Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C22⋊C4 |
| commutator | C22 | C22×C32×C4 |
| Frattini, cosocle | C2×C4 | C23×C32 |
| socle | C2×C32 | C2×C22⋊C4 |
| prime radical | C2×C32×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G | |
450 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8,11,16,12,15,10,14)(9,17,18,22,19,13,20,21);(4,5,6)(7,9)(8,13)(10,18)(11,20)(12,19)(14,21)(15,17)(16,22);(7,10,12,11)(8,14,15,16)(9,18,19,20)(13,21,17,22)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4868);