it is motivated by Density of congruence classes covered by a set

Let say just "$n$-gon" for the set of vertices of a regular $n$-gon inscribed in a unit circle, "2-gon" for the set of two opposite points.

is it true that for given positive integers $1 < b_1 < b_2 < \dots < b_k$ the union of $b_i$-gons has the minimal cardinality when they all have a common vertex?

In other words, if $G_i$ are subgroups of the finite cyclic group $G$, is it true that $$|\cup_{i=1}^k x_iG_i|\ge |\cup_{i=1}^k G_i|$$ for arbitrary cosets $x_iG_i$?