Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#3019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
762 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(6,7)(8,14)(9,17)(10,12)(11,15)(13,19)(16,21)(18,20);(6,8)(7,11)(9,18)(10,16)(12,20)(13,21)(14,17)(15,19);(9,19)(13,17)(14,21)(15,18);(7,12)(11,20)(13,17)(14,21)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18696);