Question

Maple gives Galois group $H$ of your polynomial $f=f_n$ as the subgroup of $S_8$ generated by these permutations: $$(1 2)(5 6), (1 2 3)(4 6 5), (1 2 6 3 4 5 7), (1 8)(2 3)(4 5)(6 7), (2 8)(1 3)(4 6)(5 7), (4 8)(1 5)(2 6)(3 7).$$ That can be easily proved by using the standard technique of Galois theory assuming that $n$ is such that your polynomial is irreducible (I think that the only exceptional values are 17 and 145 as in Alastair's answer). See Van der Waerden's book, for example: consider the polynomial in $9$ variables $$g(x,x_1,...,x_8)=\prod_{\sigma\in S_8} (x-x_{\sigma 1}a_1-...-x_{\sigma 8}a_8) $$ where $a_1,...,a_8$ are the roots of your polynomial $f=f_n$, $\sigma$ runs over all permutations of $S_8$. The coefficients of $g$ are symmetric polynomials in $a_i$, hence polynomials in the coefficients of $f_n$. Now show that the products of factors corresponding to permutations from $H$ form an irreducible factor of $g$. Hence $H$ is the Galois group of $f_n$ (again, assuming $f_n$ is irreducible).