Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3068 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(7,11)(8,14)(9,17)(13,19)(15,16)(18,21);(6,7,9,11,10,13,17,19)(8,12,15,18,16,20,14,21);(6,8)(7,12)(9,15)(10,16)(11,18)(13,20)(14,17)(19,21);(7,13)(11,19)(12,20)(18,21)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18745);