6. For any real numbers $x, y, z$ not all zero, we have
$$
\begin{array}{l}
-6 x y + 18 z x + 36 y z . \\
\leqslant k\left(54 x^{2} + 41 y^{2} + 9 z^{2}\right) .
\end{array}
$$

Then the minimum value of the real number $k$ is
6.1.

By the Cauchy-Schwarz inequality, we have
$$
\begin{array}{c}
{\left[(x+y)^{2}+(z+x)^{2}+(y+z)^{2}\right]\left(4^{2}+8^{2}+10^{2}\right)} \\
\geqslant[4(x+y)+8(z+x)+10(y+z)]^{2} .
\end{array}
$$

Simplifying, we get $-11r$.
$$
54 x^{2}+41 y^{2}+9 z^{2} \geqslant-6 x y+18 z x+36 y z \text {. }
$$

When $x: y: z=1: 3: 7$, the equality holds.
Thus, the minimum value of $k$ is 1.