A quasitoric manifold $M^{2n}$ over the cube $I^n$ is studied. The Stiefel-Whitney classes are calculated and used as the obstructions for immersions, embeddings and totally skew embeddings. The manifold $M^{2n}$, when $n$ is a power of 2, has interesting properties: $\operatorname{imm}(M^{2n})=4n-2$, $peratorname{em}(M^{2n})=4n-1$ and $N(M^{2n})\geq 8n-3$.