Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#1357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #331 |
| commutator | C42 | C2×C4×C5 |
| Frattini, cosocle | C4×C8 | C22×C5 |
| prime radical | C5×#493 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×C8 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7,8,13,9,12,16,20,10,14,17,21,18,11,15,19);(7,11)(8,15)(9,18)(12,14)(13,21)(16,17)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 490);