Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#397 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C23 |
| commutator | C23 | C23×C9 |
| Frattini, socle, cosocle | C23×C3 | C23×C3 |
| prime radical | C24×C3 | C22×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
255 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(14,15,16,17)(19,22)(23,24);(11,13)(18,19)(20,24)(21,22)(23,25);(10,11)(12,13)(14,15)(16,17)(18,20,21,25)(19,23,22,24)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 833);