Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#2711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C2×C5×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(11,14)(12,16)(18,19,22,21)(20,24,25,23);(7,9)(11,14)(15,17)(19,21)(20,25);(6,7)(8,9)(10,11,13,14)(12,15,16,17)(18,20,22,25)(19,23,21,24);(10,12)(11,15)(13,16)(14,17)(18,21,22,19)(20,24,25,23)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18388);