Suppose $Gamma$ is a graph with $V(Gamma) = { 1, 2,dots, p}$ and $ mathcal{F} = {Gamma_1,dots, Gamma_p} $ is a family of graphs such that $n_j = |V(Gamma_j)|$, $1 leq j leq p$. Define $Lambda = Gamma[Gamma_1,dots, Gamma_p]$ to be a graph with vertex set $ V(Lambda)=bigcup_{j=1}^pV(Gamma_j)$ and edge set $E(Lambda)=big(bigcup_{j=1}^pE(Gamma_j)big)cupbig(bigcup_{ijin E(Gamma)}{uv;uin V(Gamma_i),vin V(Gamma_j)}big) $. The graph $ Lambda$ is called the $Gamma$-join of $ mathcal{F}$. The power graph $mathcal{P}(G)$ of a group $G$ is the graph which has the group elements as vertex set and two elements are adjacent if one is a power of the other. The aim of this paper is to prove that $mathcal{P}(mathbb{Z}_{n}) = K_{phi(n)+1} + Delta_n[K_{phi(d_1)}, K_{phi(d_2)},dots, K_{phi(d_{p})}]$, where $Delta_n$ is a graph with vertex and edge sets $V(Delta_n)={d_i | 1,nnot = d_i | n, 1leq ileq p}$ and $ E(Delta_n)={ d_id_j | d_i|d_j, 1leq i<jleq p}$, respectively. As a consequence it is proved that $Aut(mathcal{P}(mathbb{Z}_{n}))cong S_{phi(n)+1}timesprod_{1,nnot=d|n}S_{phi(d)}.$ This proves a recent conjecture by Doostabadi et al. [A. Doostabadi, A. Erfanian and A. Jafarzadeh, Some results on the power graph of groups, The Extended Abstracts of the 44th Annual Iranian Mathematics Conference, 27-30 August 2013, Ferdowsi University of Mashhad, Iran]. Finally, we apply our results to obtain complete descriptions of the power graphs of some finite groups.