Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#3018 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C2×C5 | C2×#120 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
724 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)(6,7)(8,15)(9,13)(10,18)(11,14)(12,16)(17,20)(19,21);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18);(6,7)(8,12)(9,13)(10,18)(11,14)(15,19)(16,20)(17,21);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18695);