Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C29 |
| direct product | C2×C5×#518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator | C2 | C24×C4×C5 |
| Frattini, cosocle | C22 | C25×C5 |
| prime radical | C23×C5×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(15,16,17,18,19);(3,4,5,6)(7,8)(9,13)(10,11)(12,14);(8,11)(12,13);(7,8)(9,12)(10,11)(13,14);(7,9)(8,12)(10,14)(11,13)340 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21292);