Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(4,8)(5,12)(6,9)(11,16)(13,14)(15,17);(3,4,9,16,7,11,6,8)(5,10,12,17,14,18,13,15);(3,5)(4,10)(6,13)(7,14)(8,15)(9,12)(11,18)(16,17)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18742);