Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#576 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24×C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C32×#41 ◁ G | |
804 conjugacy classes of subgroups; only the 474 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 474 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8)(9,13)(10,12)(11,15)(14,18)(16,20)(17,19)(21,22);(4,5,6)(8,12)(10,18)(13,19)(14,15)(17,22)(20,21);(7,9)(8,13)(10,17)(11,16)(12,19)(14,21)(15,20)(18,22);(9,16)(13,20)(17,22)(19,21)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8259);