Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3501 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C22×C5×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2,3,4,5)(6,7)(8,12)(9,10)(11,13)(14,15,17,19)(16,20,21,18);(7,10)(11,12);(14,16,17,21)(15,18,19,20);(6,7)(8,11)(9,10)(12,13);(6,8)(7,11)(9,13)(10,12)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21368);