Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ G | |
324 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 150 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)(8,12)(9,16)(10,13)(15,20)(17,18)(19,21);(4,5,6)(7,8,13,20,11,15,10,12)(9,14,16,21,18,22,17,19);(7,9)(8,14)(10,17)(11,18)(12,19)(13,16)(15,22)(20,21)144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4927);