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In this paper, the averaging principle is studied for a class of multiscalestochastic partial differential equations driven by $\alpha$-stable process,where $\alpha\in(1,2)$. Using the technique of Poisson equation, the orders ofstrong and weak convergence are given $1-1/\alpha$ and $1-r$ for any $r\in(0,1)$ respectively. The main results extend Wiener noise considered byBr\'{e}hier in [6] and Ge et al. in [17] to $\alpha$-stable process, and thefinite dimensional case considered by Sun et al. in [39] to the infinitedimensional case.

Orders of strong and weak averaging principle for multiscale SPDEs driven by $\alpha$-stable process