GCM encryption for 256-bit and 512-bit block ciphers The Galois/Counter Mode of Operation (GCM)
This paper describe using GCM mode with 128-bit and 64-bit block cipher. Can we use this mode for 256-bit and 512-bit block cipher?
You could, however the one part that doesn't translate in an obvious manner is the Galois field representation; you would need to pick a field representation for $GF(2^{256})$ and $GF(2^{512})$, because those have not be predefined for those sizes.

Here's the issue; GCM does field multiplications internally; that is, it takes two $N$-bit vectors (where $N$ is the block size of the cipher), interprets them as field elements in $GF(2^N)$, multiplies those two elements, and those maps that result back into an $N$-bit vector.

Why is this a problem?  Well, exactly what is the mapping between bit vectors and field elements?  It turns out that there are a lot of plausible mappings; from a security standpoint, it doesn't matter (the GCM proofs all work regardless of the representation); however it matters a great deal for interoperability.

For the fields $GF(2^{128})$ and $GF(2^{64})$, that's all defined by the standard; in both cases, we use a polynomial representation, using the irreducible polynomial $x^{128}+x^7+x^2+x+1$ for $GF(2^{128})$ and the irreducible polynomial $x^{64}+x^4+x^3+x+1$ for $GF(2^{64})$ (and the bit order is backwards from what you'd expect; so the multiplicative identity (1) is represented by the bit pattern $80$ $00$ $00$ $00$ .. $00$).

All is well and good for the two block sizes specified by NIST, but you are asking about 256 and 512 bit block sizes; even if we stay with a polynomial representation (and the same backwards bit-ordering), you'd still need to select irreducible polynomials of the correct degree.

Looking through the web, I see a claim that the polynomials $x^{256}+x^{10}+x^5+x^2+1$ and $x^{512}+x^8+x^5+x^2+1$ are irreducible; I cannot confirm that (and the web site has a cavaet as well); however if so, they would be viable options.  This paper also lists some primitive (and hence irreducible) polynomials, however they aren't minimal (which would be the obvious one to use)