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The projective special orthogonal group $PSO^+_8( 3) cong O^+_8(3).2_1$ is a group obtained from the special orthogonal group $SO_8(3)$ on factoring by the group of scalar matrices it contains. The group $PSO^+_8( 3)$ has a maximal subgroup of the form $2^6{cdot}S_8$ with index $3838185$. The group $ar{Q}cong 2^6{cdot}S_8$ is a non-split group extension of an elemenrary abelian $2$-group of order $64$ by the symmetric group $S_8$. In the present article, we construct the non-split extension $ar{Q}$ and compute its character table by using the Fischer-Clifford matrices.