  We study the S=1/2 Heisenberg antiferromagnet on the Cairo pentagon lattice
by the numerical-diagonalization method. We tune the ratio of two
antiferromagnetic interactions coming from two kinds of inequivalent sites in
this lattice, examining the magnetization process of the antiferromagnet;
particular attention is given to one-third of the height of the saturation. We
find that quantum phase transition occurs at a specific ratio and that a
magnetization plateau appears in the vicinity of the transition. The plateau is
accompanied by a magnetization jump on one side among the edges due to the
spin-flop phenomenon. Which side the jump appears depends on the ratio.
