  We introduce the notion of the power quandle of a group, an algebraic
structure that forgets the multiplication but keeps the conjugation and the
power maps. Compared with plain quandles, power quandles are much better
invariants of groups. We show that they determine the central quotient of any
group and the center of any finite group. Any group can be canonically
approximated by the associated group of its power quandle, which we show to be
a central extension, with a universal property, and a computable kernel. This
allows us to present any group as a quotient of a group with a
power-conjugation presentation by an abelian subgroup that is determined by the
power quandle and low-dimensional homological invariants.
