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×#3485 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1560 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)(10,11)(12,18)(13,21)(14,15)(16,20)(17,22)(19,23)(24,25);(11,15)(16,18)(17,21)(23,24);(6,7)(8,9)(10,12,14,20)(11,16,15,18)(13,19,22,25)(17,23,21,24);(10,11)(12,16)(13,17)(14,15)(18,20)(19,23)(21,22)(24,25);(10,13)(11,17)(12,19)(14,22)(15,21)(16,23)(18,24)(20,25)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21352);