Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C22×C5×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C25×C5 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1320 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)(21,22,23,24,25);(3,4);(6,10)(7,14)(12,19)(13,17)(15,18)(16,20);(5,6,11,17)(7,12,18,19)(8,10,9,13)(14,20,15,16);(5,7,8,15)(6,12,10,16)(9,14,11,18)(13,20,17,19)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21305);