Let $G = \Gamma_0(N)$, $N \not\equiv 3 \pmod{4}$ and $g$ be the group generated by the involution $z \mapsto -1/Nz$ of the upper half plane. We determine the cohomology set $H^1(g,G)$ in terms of the class number of quadratic forms of discriminant $-4N$.

We show that every solution of the first Painlevé equation has the finite growth order. The second and the fourth Painlevé equations have the same property.