jmc

We can write $$\begin{array}{rl}{x}^{512}+x^{256}+1& =(x^{512}-x^2)+(x^{256}-x)+(x^2+x+1)\\ & =x^2(x^{510}-1)+x(x^{255}-1)+(x^2+x+1)\\ & =x^2(x^3-1)(x^{507}+x^{504}+x^{501}+\cdots +x^3+1)\\ & +x(x^3-1)(x^{252}+x^{249}+x^{246}+\cdots +x^3+1)\\ & +x^2+x+1\\ & =(x-1)(x^2+x+1)(x^{509}+x^{506}+x^{503}+\cdots +x^5+x^2)\\ & +(x-1)(x^2+x+1)(x^{253}+x^{250}+x^{247}+\cdots +x^4+x)\\ & +x^2+x+1\\ & =(x^2+x+1)(x^{510}-x^{509}+x^{507}-x^{506}+x^{504}-x^{503}+\cdots +x^6-x^5+x^3-x^2)\\ & +(x^2+x+1)(x^{254}-x^{253}+x^{251}-x^{250}+x^{248}-x^{247}+\cdots +x^5-x^4+x^2-x)\\ & +x^2+x+1.\end{array}$$Thus, $$\begin{array}{rl}P(x)& =(x^{510}-x^{509}+x^{507}-x^{506}+x^{504}-x^{503}+\cdots +x^6-x^5+x^3-x^2)\\ & +(x^{254}-x^{253}+x^{251}-x^{250}+x^{248}-x^{247}+\cdots +x^5-x^4+x^2-x)+1\\ & =x^{510}-x^{509}+x^{507}-x^{506}+\cdots +x^{258}-x^{257}\\ & +x^{255}-x^{254}+x^{252}-x^{251}+\cdots +x^3-x^2\\ & +x^{254}-x^{253}+x^{251}-x^{250}+\cdots +x^2-x+1\\ & =x^{510}-x^{509}+x^{507}-x^{506}+\cdots +x^{258}-x^{257}\\ & +x^{255}-x^{253}+x^{252}-x^{250}+\cdots +x^3-x+1.\end{array}$$Among $x^{510},$ $-x^{509},$ $x^{507},$ $-x^{506},$ $\dots ,$ $x^{258},$ $-x^{257},$ there are 170 nonzero coefficients.

Among $x^{255},$ $-x^{253},$ $x^{252},$ $-x^{250},$ $\dots ,$ $x^3,$ $-x,$ there are another 170 nonzero coefficients.

The final term of 1 gives us a total of $\boxed{341}$ nonzero coefficients.