In most images of $1s$ subshell I see that there's no node shown at the center, and even the formula $n-\ell-1$ gives 0 as the answer.
But, isn't the nucleus experimentally proven to be at the center? And as the mass should be much more than individual electrons, I'm assuming the nucleus doesn't have large probability clouds like individual electrons do - i.e. the nucleus is mostly at the center.
If my assumption is correct, then shouldn't it be impossible for the electron cloud to occupy any space at the center as it's already taken up? Even for the other orbitals there seems to be empty space in the center, so why not for the $s$ orbital?
I did see one image of the $s$ orbitals where it appeared like they showed a node but that was just one image, every text or any other image says otherwise. So, could someone verify and explain whether there is a node or not?