Growth of Au-adsorbed 5$\ifmmode\times\else\texttimes\fi{}$2 domains on a vicinal Si(111) surface misoriented toward the [11$\overline{2}$] direction is studied by scanning tunneling microscopy (STM). We find that, at a coverage of 0.3 ML, the 5$\ifmmode\times\else\texttimes\fi{}$2 domains are elongated in the [0$\overline{1}$1] or the $[\overline{1}$01] direction, not parallel to the step edge. The boundary to the Si(111)-7$\ifmmode\times\else\texttimes\fi{}$7 domain shows a characteristic structure depending on whether it faces faulted or unfaulted halves. Within the 5$\ifmmode\times\else\texttimes\fi{}$2 domains, several rows of bright protrusions run parallel to the boundaries. We find that the domain width ${d}_{\mathrm{AB}}$ in the $[1\overline{1}0]$ direction is expressed as ${d}_{\mathrm{AB}}{=5N}_{\mathrm{rows}}+5$, where the ${N}_{\mathrm{rows}}$ is the number of rows in the domain, and the length is represented in units of the lattice constant of the 1$\ifmmode\times\else\texttimes\fi{}$1 surface. Moreover, we observe that phase boundaries of the 7$\ifmmode\times\else\texttimes\fi{}$7 surface are often terminated by the 5$\ifmmode\times\else\texttimes\fi{}$2 domains. Then, the component of the Burgers vector along the $[1\overline{1}0]$ direction determines the domain width ${d}_{\mathrm{AB}}$ and the number of rows ${N}_{\mathrm{rows}},$ and is found to be the remainder of ${(d}_{\mathrm{AB}}+5)$ divided by 7. These facts mean that the values of ${d}_{\mathrm{AB}}$ are quantized; these are discrete at intervals of 5, and are increased by 35: the number 35 is the least common multiple of the periodicities of the 7$\ifmmode\times\else\texttimes\fi{}$7 and the 5$\ifmmode\times\else\texttimes\fi{}$2 structure in the $[1\overline{1}0]$ direction. We propose a growth process of the 5$\ifmmode\times\else\texttimes\fi{}$2 domain based on the STM observation.