Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#2330 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C24 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C23 | C24×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,16)(13,14)(15,17)(19,21)(20,24);(6,8,9,7)(11,14)(15,16)(18,19,22,21)(20,25,24,23);(10,11)(12,15)(13,14)(16,17)(18,20,22,24)(19,23,21,25);(10,12,13,17)(11,15,14,16)(18,21,22,19)(20,25,24,23)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18007);