Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3480 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| socle | C22×C5 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2,3,4,5)(7,10)(11,13)(14,15,18,20)(16,19,17,21);(6,7)(8,11)(9,10)(12,13)(14,16,18,17)(15,19,20,21);(14,17,18,16)(15,19,20,21);(7,10)(11,13);(6,8,9,12)(7,11,10,13)250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21347);