Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#471 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #121 |
| commutator | C2×C4 | C23×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#121 |
| prime radical | C3×AGL(1,ℤ/8) | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ G | |
156 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(11,15)(12,18)(13,21)(17,23)(19,20)(22,24);(10,11,13,15,14,17,21,23)(12,16,19,22,20,25,18,24);(10,12)(11,16)(13,19)(14,20)(15,22)(17,25)(18,21)(23,24)144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 907);