The presentation for $\pi_1(X)$ that you write down simplifies to $\langle b, e \mid beb=eb^2e, ebe=be^2b\rangle$. As is well-known, the binary icosahedral group $I^*$ is isomorphic to the group of unimodular, $2\times 2$ matrices over $\mathbb{Z}_5$. An explicit isomorphism $\pi_1(M) \to {\rm SL}(2,5)$ is given by $$b \mapsto \begin{pmatrix} 3 & 1 \\ -1 & 0 \end{pmatrix}, \quad e \mapsto \begin{pmatrix} -1 & -1 \\ 0 & -1 \end{pmatrix}.$$
The presentation for $\pi_1(X)$ that you write down simplifies to $\langle b, e \mid beb=eb^2e, ebe=be^2b\rangle$. As is well-known, the binary icosahedral group $I^*$ is isomorphic to the group of unimodular, $2\times 2$ matrices over $\mathbb{Z}_5$. An explicit isomorphism $\pi_1(M) \to {\rm SL}(2,5)$ is given by $$b \mapsto \begin{pmatrix} 3 & 1 \ -1 & 0 \end{pmatrix}, \quad e \mapsto \begin{pmatrix} -1 & -1 \ 0 & -1 \end{pmatrix}.$$