Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#537 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C24 |
| commutator | C22 | C24×C9 |
| Frattini, socle, cosocle | C22×C3 | C24×C3 |
| prime radical | C3×#537 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
441 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15)(16,20)(17,18)(19,21);(15,18)(19,20);(10,11)(12,13)(14,15)(16,19)(17,18)(20,21);(14,16,17,21)(15,19,18,20)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1851);