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×#3514 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1142 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)(8,13)(9,12)(10,11)(15,18)(19,21);(7,11)(12,13);(6,8,10,9)(7,12,11,13)(14,15)(16,19)(17,18)(20,21);(6,7)(8,12)(9,13)(10,11);(6,9,10,8)(7,13,11,12)(14,16,17,20)(15,19,18,21)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21381);