If all proper factors of a word u are beta-power-free while u itself is not, then u is a minimal beta-power. We consider the following general problem: for which numbers k, beta, and p there exists a k-ary minimal beta-power of period p? For the case beta >= 2 we completely solve this problem. If the number beta < 2 is relatively "big" w.r.t. k, we show that any number p can be the period of a minimal beta-power. Finally, for "small" beta we describe some sets of forbidden periods and provide a numerical evidence that for k >= 9 these sets are almost exhaustive.