Let A be a $3 \times 3$ matrix such that ${X^T}AX = O$ for all nonzero $3 \times 1$ matrices $X = \left[ {\begin{array}{*{20}{l}} x\\ y\\ z \end{array}} \right]$. If ${\rm{A}}\left[ {\begin{array}{*{20}{l}} 1\\ 1\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ 4\\ { - 5} \end{array}} \right],{\rm{A}}\left[ {\begin{array}{*{20}{l}} 1\\ 2\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 4\\ { - 8} \end{array}} \right]$ , and $\det ({\mathop{\rm adj}\nolimits} (2(\;{\rm{A}} + {\rm{I}}))) = {2^\alpha }{3^\beta }{5^\gamma },\alpha ,\beta ,\gamma \in $, then ${\alpha ^2} + {\beta ^2} + {\gamma ^2}$ is