Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#1304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #95 |
| commutator | C23 | C42×C5 |
| Frattini, cosocle | C25 | C22×C5 |
| prime radical | C25×C5 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C25×C5 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7)(8,14,16,19)(9,12)(10,13)(11,15,18,21)(17,20);(6,8)(7,11,13,19)(9,16)(10,15)(12,18,20,14)(17,21)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 437);