Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3483 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| socle | C22×C5 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1202 conjugacy classes of subgroups; only the 800 normal subgroups are shown.
This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 800 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) = 320 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2,3,4,5)(7,11)(12,13)(14,15)(16,19)(17,21)(18,20);(6,7,10,11)(8,12,9,13)(14,16,19,15)(17,21,18,20);(6,8,10,9)(7,12,11,13)(14,17,19,18)(15,20,16,21);(6,8,10,9)(7,13,11,12)(14,18,19,17)(15,20,16,21);(6,9,10,8)(7,13,11,12)250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21350);