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×#3499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11,14,17)(12,15,20,18)(13,21,22,16)(19,24,25,23);(10,11)(12,18)(13,16)(14,17)(15,20)(19,24)(21,22)(23,25);(6,7,8,9);(10,12)(11,15)(13,19)(14,20)(16,23)(17,18)(21,24)(22,25);(10,13,14,22)(11,16,17,21)(12,19,20,25)(15,23,18,24)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21366);