Matrix multiplication based on Strassen's algorithm is in $O(n^{\log(7)/\log(2)})$ and is quite practical. As far as I am aware, for any exponent $\omega<\log(7)/\log(2)$ the corresponding algorithm is impractical, indeed because of huge constants.

Matrix multiplication based on Strassen's algorithm is in $O(n^{\log(7)/\log(2)})$ and is quite practical. As far as I am aware, for any exponent $\omega<\log(7)/\log(2)$ the corresponding algorithm is impractical, indeed because of huge constants.

Added Oct. 9, 2022:

Apparently, Alphatensor by Deepmind has found (many) ways to multiply $4\times4$ matrices in $47$ multiplications, so I guess the "practicality exponent" is now down to $\log(47)/\log(4)$.