Algebraicity of a ratio of values of the Gamma function

This number is expected to be transcendental. This answer gives a conceptual framework for studying the algebraicity of such $Gamma$ ratios, and in fact a completely explicit criterion (which is only conjectural, when it comes to establishing transcendence).

Your number is equal to begin{equation*} frac{Gamma(2/5)^3}{Gamma(1/5)^2 Gamma(4/5)} end{equation*} up to multiplication by an algebraic number. This ratio is described by the vector of exponents $(-2,3,0,-1)$ and the criterion there is not met for $u=2$. In fancy terms, the Bernoulli distribution $B_1$ does not vanish on this vector.