Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
756 conjugacy classes of subgroups; only the 426 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 426 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 — 4 generators on 22 points:
(1,2,3)(7,8)(9,16)(10,15)(11,13)(12,18)(14,20)(17,21)(19,22);(4,5,6)(7,9)(8,12)(10,18)(11,19)(13,21)(14,22)(15,16)(17,20);(8,13)(9,17)(12,16)(14,15)(18,19)(21,22);(8,14)(9,18)(12,21)(13,15)(16,22)(17,19)171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8243);