xy The problem of finding upper bounds for minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic \(2\)-groups. This problem was considered earlier by other authors. We give a construction of an undirected graph having \(2^{n}+6\) vertices and automorphism group cyclic of order \(2^{n}\), \(n\geq 2\). This can revive interest in related problems.