On min-base palindromic representations of powers of 2

A positive integer $N$ is \emph{palindromic in the base $b$} when $N={\sum }_{i=0}^k{c}_i{b}^i$, $c_k\mathrm{\ne }0$,and $c_i=c_{k-i},i=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.},k$, Focusing on powers of 2, we investigate the smallest base $b$ when $N=2^n$ is palindromic in the base $b$.