What constant in the latest fast matrix multiplication is hidden by Big-O notation?
All improvements since Strassen's original algorithm only achieve their exponent in the limit. That is, if they are stated as $O(n^c)$ algorithms, what they really show is that for every $\epsilon&gt;0$ you can multiply two $n\times n$ matrices using at most $C_\epsilon n^{c-\epsilon}$ multiplications, where the constant $C_\epsilon$ blows up badly (exponentially in $1/\epsilon$?) as $\epsilon\to0$. (Coppersmith and Winograd showed that this kind of behavior is actually necessary.)

The papers don't explicitly compute $C_\epsilon$. One reason is that if you are serious about the constant, then you will probably need to apply low-level optimizations that will improve $C_\epsilon$ at the expense of the simplicity of the algorithm. Another reason is that due to the exponential dependence on $1/\epsilon$, you don't expect the result to be practically meaningful, so it's not really worth it. A third reason is that fast matrix multiplication algorithms are developed in a more abstract framework, which makes it awkward to come up with an actual explicit algorithm.

Fast matrix multiplication algorithms are recursive, that is, they iterate a smart $m\times m$ multiplication algorithm. Whereas for Strassen's original algorithm $m = 2$, for the improvements $m$ depends on $\epsilon$ (exponentially?), and so you will only be able to apply the base case for very large matrices; and for such large matrices, you would actually do much worse than the high school algorithm. The benefits of these algorithms only show up once they have been iterated many times, which would correspond to astronomically high (or worse) dimensions.

If you do feel like working out a constant, there are several expositions of the theory of fast matrix multiplication. You could take the first improvement over Strassen's original algorithm, and try to work out what the parameters are like.