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 | C2×C5×#553 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4)(5,6)(7,8)(9,10)(12,15)(16,17);(4,8)(6,9)(11,12,14,15)(13,17,18,16);(3,5)(4,9)(6,8)(7,10);(3,6,7,9)(4,5,8,10)(11,13,14,18)(12,16,15,17)250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21338);