Growth of Au-adsorbed 5 X 2 domains on a vicinal Si(111) surface misoriented toward the [11(2) over bar 2] direction is studied by scanning tunneling microscopy (STM). We find that, at a coverage of 0.3 ML, the 5 X 2 domains are elongated in the [0(1) over bar1$] or the [(1) over bar 01] direction, not parallel to the step edge. The boundary to the Si(111)-7 X 7 domain shows a characteristic structure depending on whether it faces faulted or unfaulted halves. Within the 5 X 2 domains, several rows of bright protrusions run parallel to the boundaries. We find that the domain width d(AB) in the [1(1) over bar0$] direction is expressed as d(AB) = 5N(rows) + 5, where the N-rows is the number of rows in the domain, and the length is represented in units of the lattice constant of the 1 X 1 surface. Moreover, we observe that phase boundaries of the 7 X 7 surface are often terminated by the 5 X 2 domains. Then, the component of the Burgers vector along the [1(1) over bar0$] direction determines the domain width d(AB) and the number of rows N-rows, and is found to be the remainder of (d(AB) + 5) divided by 7. These facts mean that the values of d(AB) are quantized; these are discrete at intervals of 5, and are increased by 35: the number 35 is the least common multiple of the periodicities of the 7 X 7 and the 5 X 2 structure in the [1(1) over bar0$] direction. We propose a growth process of the 5 X 2 domain based on the STM observation.