Let $G$ be a (topological) group whose identity element $e_G$ is a nondegenerated basepoint (e.g. if $G$ is a Lie group). Then that's a known fact that there is for every 'nice' enough topological space $X$ (eg paracompact sp, or more elementary a CW complex) a natural bijection

$$ [X, BG] \cong \mathcal{P}_G(X) $$

between homotopy classes of cont maps $X \to BG$ and
isomorphism classes of of principal $G$-bundles over base $X$. Explicitely
this is given by associating to a homotopy class of a map
$f: X \to BG$ the pullback bundle $f^*EG$ of the *universal*
principal $G$-bundle $u:EG \to BG$.

Now like the trivial bundle the universal bundle $EG$ is the second
'canonical' (up to isomorphism) $G$-bundle and besides it's universal
property I'm trying to understand if it's possible to deduce it's
universal property *only* from study of it's *$1$-cocycle* represenation. Let's recall
that we can over every nice enough base space $X$ encode the complete information
about a $G$-bundle $E \to X$ with fiber $F$ (therefore $G \subset Aut(F)$;
in our case later it will be $G$ itself since we are interested on
principal bundles) in terms of a *$1$-cocycle*. Let $\mathcal{U}= \{U_i \}_i$ covering of $X$ over which $E$ trivializes, then

(1) a colection of homeomorphisms $\psi_i: U_i \times F \to p^{-1}(U_i)$ compatible with projection $p$

(2) the *$1$-cycle* (ie the set of *transition functions*) $g_{ij}: U_j \cap U_i \to G$
such that for every $(x, f) \in (U_j \cap U_i) \times F$ we have

$$ \psi_j^{-1} \circ \psi_i (x,f) = (x, (g_{ji}(x))(f))$$

(3) the transition functions satisfy $g_{ii} = e_G$ (constant function), $g_{ji}= g_{ij}^{-1}$ and $g_{kj} \cdot g_{ji}= g_{ki}$

Moreover two $G$-bundles $E_1, E_2$ over same base $X$ are isomorphic if there exist fine enough covering $\{U_i \}_i$ of $X$ such that there eist a family $g_i: U_i \to G$ with

$$ g^2_{ij} = g_i \cdot g^1_{ij} \cdot g_j^{-1} $$

where $\{g^1_{ij}\}$ and $\{g^2_{ij}\}$ are $1$-cycles of $E_1$ and $E_2$.

* Question*: Is it known which $1$-cocycle $\{g^{EG}_{ij}\}$ is naturally associated
to the universal bundle $u:EG \to BG$? Is there any canonical choice
of the covering $\{U_i \}_i$ of $BG$ known over which this
$1$-cocycle can be in most transparent way be declared? Is it known at least if we work with a

*finite group*? The model of $EG$ and $BG$ I have here in mind is that one by Milnor explaned in detail for example here.

If we work with this model for $EG \to BG$ can it's $1$-cocycle be writen down? It must be something canonical but I don't know how to determine it. My motivation is to understand if it's possible to understand only by study the structure of it's $1$-cocycle that this $G$-bundle is in certain way the most 'flexible' one in the sense such that every other can be derived up to iso by the pullback of it, as the universal property says. So speaking informally every 'twist' of $G$-bundle should be already 'known' in some way already to the universal bundle, such it 'knows' all possible twists can happen only by taking pullbacks. (Ok, I admit, that the last formulation is a bit vague, but that's the core of my motivation: I understand 'formally' the universal property of the universal bundle, but have no intuition on it's intrinsical geometry why and why exacly this geometry is the most 'flexible' to allow to recover all over $G$-bundles by taking pullbacks of it.