Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2839 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,10)(11,13)(14,15)(16,18)(17,19)(20,21)(22,23,26,28)(24,25,27,29);(8,12)(11,13)(14,15,17,19)(16,20,21,18)(22,24)(23,25)(26,27)(28,29);(6,7,9,10)(8,11,12,13)(14,15)(16,20)(17,19)(18,21)(22,25)(23,27)(24,28)(26,29);(6,8)(7,11)(9,12)(10,13)(14,16,17,21)(15,18,19,20)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18516);