Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C32 |
| socle | C2×C32 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G | |
786 conjugacy classes of subgroups; only the 474 normal subgroups are shown.
This group has 786 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)(9,16)(10,15)(11,19)(12,14)(13,17)(18,22)(20,21);(4,5,6)(7,9)(8,13)(10,17)(11,20)(12,18)(14,21)(15,22)(16,19);(8,14)(13,21)(15,19)(16,22);(7,10,12,11)(8,15,14,19)(9,17,18,20)(13,22,21,16)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8258);