Analyzing security flaws of cipher
When seeing clearly through all the cube "magic", one recognizes the following: All the cube operations are just key-dependent bit permutations. Therefore, the whole cipher is a sequence of key-dependent permutations and XORs with key bits.

This admits an algebraic description: For all keys, there is a permutation matrix $A\in\mathbb F_2^{512\times512}$ and a vector $b\in\mathbb F_2^{512}$ such that
$$\mathit{cipher} = A\cdot \mathit{plain}+b\text,$$
where $\mathit{plain}$ and $\mathit{cipher}$ are interpreted as vectors over $\mathbb F_2$ of length $512$. Therefore, the cipher is an affine map over $\mathbb F_2$. This is about the most undesirable property a block cipher can exhibit. An attacker has of course no direct access to $A$ and $b$, but there are lots of weaknesses: For example, in a chosen-plaintext scenario, an attacker can


encrypt $\mathit{plain}=0$, thus obtaining $\mathit{cipher}=b$; and subsequently
encrypt $\mathit{plain}=\mathrm e_i$, the $i$th unit vector, for all $i$. He can then subtract $b$ from the resulting ciphertext, yielding $A\cdot\mathit{plain}$, and check which entry of this vector is $1$ — this is the target of the $i$th entry of $\mathit{plain}$ under the permutation $A$.


A chosen-plaintext attacker is therefore capable of obtaining (information equivalent to) the encryption key using a very simple procedure, hence making the cipher, to put it mildly, rather unsatisfactory (which is a synonym for "absolutely broken beyond any hope" in cryptography).