Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C5 |
| prime radical | C5×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ G | |
246 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 21 points:
(1,2,3,4,5)(6,7)(8,16)(9,13)(10,14)(11,15)(12,17)(18,20)(19,21);(6,8,11,9)(7,12,15,20)(10,17,19,18)(13,14,16,21);(6,9,11,8)(7,13,15,16)(10,17,19,18)(12,14,20,21)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5788);