Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#3487 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4,5)(7,9)(11,12)(14,22)(16,18)(20,25);(10,11,15,12)(13,18,21,16)(14,17,22,19)(20,24,25,23);(6,7)(8,9)(10,12,15,11)(13,18,21,16)(14,19,22,17)(20,24,25,23);(10,13,15,21)(11,16,12,18)(14,20,22,25)(17,23,19,24);(10,14)(11,17)(12,19)(13,20)(15,22)(16,23)(18,24)(21,25)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21354);