TITLE: Is there a natural family of nonisomorphic groups parametrized by $\mathbb{R}$?
QUESTION [10 upvotes]: It's easy to construct a countable series of distinct groups - the cyclic groups, for instance - and it's also easy to create a family of groups parametrized by the reals, but most such constructions will have isomorphisms betwteen most if not all of the groups.
As the category Group is very large, one would expect that somewhere inside it lies a continuum-sized collection of groups which can be parametrized in some natural way and are all distinct, that is, nonisomorphic. However, I've pondered this for a bit and asked some friends and haven't come up with any clear constructions. So I'm curious if anyone can provide a collection of groups which is "natural" in some sense (ideally continuous, and not a composition of a bijection from a set of groups to $\{0,1\}^{\mathbb{N}}$ and from there to $\mathbb{R}$), continuum-sized, and whose elements are not isomorphic. 
(Answers which provide a family parametrized by the nonnegative reals, or the positive reals, or the interval $[0,1]$, would also suffice.)

REPLY [3 votes]: As proved in this question, the groups $G_{x^r}$ for $r\in (0,1]$ (defined as permutations of $\mathbb{N}$ that fix all but  $o(n^r)$ elements) are pairwise non-isomorphic.