Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#2398 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| socle | C23×C5 | C24 |
| prime radical | C22×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(10,11,14,16)(12,17,13,15)(18,19,21,23)(20,24,25,22);(6,7)(8,9)(10,12,14,13)(11,15,16,17)(18,20,21,25)(19,22,23,24);(7,9)(19,23)(22,24);(10,13,14,12)(11,15,16,17)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18075);