Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#2370 |
| 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 | |
464 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)(10,11,14,13)(12,16,17,15)(18,19)(20,24)(21,25)(22,23);(6,8,9,7)(11,13)(15,16)(19,23)(21,24);(10,12)(11,15)(13,16)(14,17)(18,20,22,25)(19,24,23,21);(10,13,14,11)(12,16,17,15)(18,21,22,24)(19,25,23,20)250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18047);