The irreducible but solvable octic,
$$x^8-x^7+29x^2+29=0\tag{1}\label{1}$$
was first mentioned by Igor Schein in this 1999 sci.math post (Wayback Machine). This does not factor over a quadratic or quartic extension, but over a 7th deg one. It can also be nicely solved using the $\color{blue}{29}$th root of unity. Let $\omega = \exp(2\pi i /29)$ then define,
\begin{gather} y_k = \omega^{k}+\omega^{12k}+\omega^{17k}+\omega^{28k}\tag{2}\label{2} \\ z_k = 4(y_k^3+y_k^2-9y_k-4)(y_k^2-2)(y_k-1)+9\tag{3}\label{3} \end{gather}
then I found a pair of octic roots as,
\begin{gather*} x = \frac{1\color{red}{-}\sqrt{z_{1}}+\sqrt{z_{2}}+\sqrt{z_{4}}+\sqrt{z_{8}}+\sqrt{z_{16}}+\sqrt{z_{32}}+\sqrt{z_{64}}}{8} \approx 1.79106+0.8286\,i\dots \\ x = \frac{1+\sqrt{z_{1}}\color{red}{-}\sqrt{z_{2}}\color{red}{-}\sqrt{z_{4}}+\sqrt{z_{8}}+\sqrt{z_{16}}\color{red}{-}\sqrt{z_{32}}+\sqrt{z_{64}}}{8} \approx 1.79106-0.8286\,i\dots \end{gather*}
and the other pairs using appropriate signs of the square roots. Note that the seven $y_k$ and $z_k$ are roots of two different 7th-deg eqns with integer coefficients, with the $z_k$ being,
$$\small{z^7 - 7z^6 - 2763z^5 - 19523 z^4 + 1946979z^3 + 34928043 z^2 + 119557031z - 3247^2=0}$$
and \eqref{3} is the 6th-deg Tschirnhausen transformation between the $y_k$ and $z_k$. (In an earlier edit, I used an alternative expression for $z_k$ by P. Montgomery found in the sci.math link, but I like this one better.)
Question: Does anyone know why \eqref{1} has such a simple form, and if we can find other irreducible but solvable octics with the same Galois group involving a $p$th root of unity for other prime $p$? (For some reason, this does not appear in the Kluener's database of number fields for 8T25.)