I cannot see a way to visualize it graphically as it is asked in the question.
It is not necessary nor required of you to visualize or draw a wanted dfa. The exercise only asks you to "construct a DFA accepting ...". The meaning of constructing a DFA is specifying/describing/defining the 5-tuple, $(Q,\Sigma, \delta, q_0,F)$ in the definition of a DFA. Or whatever definition that is used in your course. If you can draw a graph of the DFA, that is great. However, you don't have to unless you are asked specifically to include a graph.
Let us describe a desired DFA as follows.
- The states $Q=\{q_0, q_1, \cdots, q_{1023}\}$.
- The alphabet $\Sigma=\{0, 1\}$.
- The transition $\delta(q_i, \sigma)=q_{(i\%512<<1) + \sigma}$ for any state $q_i$ and any $\sigma\in\Sigma$. Or, what is equivalent, $$\delta(q_{(d_9d_8d_7d_6d_5d_4d_3d_2d_1d_0)_2},\sigma)=q_{(d_8d_7d_6d_5d_4d_3d_2d_1d_0\sigma)_2}$$ where $(d_9d_8d_7d_6d_5d_4d_3d_2d_1d_0)_2$ is the binary representation of the subscript, with some leading $0$s possibly.
- The initial state is $q_0$.
- The accept states are $q_{512}, q_{513}, \cdots, q_{1023}$. Put it in another way, a state $q_i$ is an accept state iff the binary representation of $i$ has $10$ digits, with the most significant digit being $1$.
It is straightforward to check the DFA constructed above accepts all strings whose 10th symbol from the right end is $1$ and no other strings.
By the way, the DFA above that has $2^{10}$ states is the minimum DFA wanted, as you have noted. It is a nice example that indicates that the minimum DFA that is equivalent to an $n$-state NFA may require $2^n$ states.