Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1940 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C23 |
| commutator | C23 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle, prime radical | C23×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
248 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 640 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(6,7)(8,9)(11,13)(14,15,19,16,18,20,17,21);(7,9)(14,16,19,20,18,21,17,15);(10,11,12,13)(14,17,18,19)(15,16,20,21)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5530);