Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C32 |
| prime radical | C32×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C2×C32×C8 ◁ G | |
246 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(7,8)(9,17)(10,21)(11,26)(12,14)(13,18)(15,24)(16,23)(19,37)(20,34)(22,32)(25,33)(27,31)(28,36)(29,38)(30,35);(4,5,6)(7,9,12,20)(8,13,14,28)(10,22,25,38)(11,27,23,19)(15,31,30,37)(16,32,26,29)(17,33,34,21)(18,35,36,24);(8,14)(13,28)(15,30)(16,26)(17,34)(21,33)(29,32)(31,37)144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4965);