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×#3021 |
| 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 | |
736 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 736 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)(7,12)(9,10)(13,20)(14,17)(15,18)(19,21);(6,7)(8,13)(9,12)(10,17)(11,14)(15,19)(16,21)(18,20);(6,8)(7,13)(9,15)(10,18)(11,16)(12,19)(14,21)(17,20);(6,9,11,10)(7,12,14,17)(8,15,16,18)(13,19,21,20)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18698);