Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C4×C5×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C23 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C23 | C24×C5 |
| socle, prime radical | C23×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4)(17,18,19,20,21);(6,8)(9,10,13,16)(11,15,12,14);(9,11,13,12)(10,14,16,15);(5,6,7,8)(9,12,13,11)(10,15,16,14)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 17984);