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×#3508 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1232 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)(16,20)(17,19)(18,21);(7,10)(11,12)(14,15,18,21)(16,19,17,20);(14,16,18,17)(15,19,21,20);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,20)(18,21);(6,8,9,13)(7,11,10,12)(14,17,18,16)(15,20,21,19)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21375);