Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#578 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C22×D4 |
| commutator | C4 | C24×C9 |
| Frattini, cosocle | C3×C4 | C24×C3 |
| socle | C2×C3 | C22×C3×D4 |
| prime radical | C3×#578 | C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C9×#41 ◁ G | |
372 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,15,19)(12,20,23,33)(13,17,24,31)(14,26,28,18)(16,27,30,22)(21,34,35,41)(25,38,37,32)(29,39,40,36);(10,12,15,23)(11,16,19,30)(13,21,24,35)(14,27,28,22)(17,29,31,40)(18,33,26,20)(25,39,37,36)(32,41,38,34);(10,13)(11,17)(12,21)(14,25)(15,24)(16,29)(18,32)(19,31)(20,34)(22,36)(23,35)(26,38)(27,39)(28,37)(30,40)(33,41);(13,24)(17,31)(21,35)(25,37)(29,40)(32,38)(34,41)(36,39)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1892);