The study of the pentagon equation leads to results on the structure and classification of finite quantum groups. It is proved that $L$ is a finite-dimensional Hopf algebra if and only if there exists an invertible matrix $R$, solution of the pentagon equation $R^{12}R^{13}R^{23}=R^{23}R^{12}$, such that $L\cong P(n, R)$; the Hopf algebra structure of $P(n, R)$ is explicitly described using generators and relations. Finally, it is proved that there exists a one-to-one correspondence between the set of types of $n$-dimensional Hopf algebras and the set of orbits of the action ${\rm GL}_n(k)\times (M_n(k)\otimes M_n(k)) \,{\longrightarrow}\, M_n(k)\otimes M_n(k), (u, R)\,{\longrightarrow}\, (u\otimes u)R(u\otimes u)^{-1}$, the representatives of which are invertible solutions of length $n$ for the pentagon equation.