Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#1504 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| prime radical | C5×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
334 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 96 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)(7,11,13,20)(8,15)(12,19,14,16)(17,21);(6,7)(8,16)(9,12)(10,14)(11,17)(13,18)(15,20)(19,21);(7,12)(11,19)(13,14)(16,20)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5094);