TITLE: Explicit constructions of K(G,2)?
QUESTION [17 upvotes]: Recall that an Eilenberg-Maclane space $K(G, n)$ is characterized by $\pi_i(K(G,n)) = G$ if $i=n$ and is trivial otherwise.  (Of course $G$ should be abelian if $n>1$.)
Let $G$ be a finite abelian group.
Below I describe cell complexes $X_1$ and $X_2$ with $\pi_2(X_i) = G$ and $\pi_0(X_i)$ and $\pi_1(X_i)$ both trivial.  By standard results it is possible to add 4-cells to $X_i$ to kill off $\pi_3$, then add 5-cells to kill off $\pi_4$, and so on.
My questions:
(1.i) Does there exist in the literature an explicit description of the 4- and 5-cells one would need to add to $X_i$ in order to turn it into a $K(G,2)$?  (I'm only interested in dimensions 4 and 5, not higher.)
(2) More generally, are there explicit descriptions of $K(G,2)$ in the literature?  (I'm already aware of making $K(G, 1)$ into a group and then applying the bar construction.)
Definition of $X_1$: A single 0-cell.  A 2-cell $c_g$ for each element $g\in G$.  A 3-cell $d_{g,h}$ for each $(g,h)\in G\times G$, with $\partial d_{g,h} = c_g + c_h - c_{gh}$.  
(This starts out similarly to a standard construction of $K(G, 1)$, but the higher dimensional cells will necessarily be more complicated.  Obvious candidates for the boundaries of 4-cells would include $d_{g,h} - d_{fg,h} + d_{f,gh} - d_{f,g}$ for all $(f,g,h)\in G\times G\times G$, and also Hopf maps to the 2-cells $c_g$ for each $g$, and also $d_{g,h} - d_{h,g} + x$, where $x$ is a map to $c_h\cup c_h$ which exhibits the commutativity of $\pi_2(c_g\cup c_h)$.)
Definition of $X_2$: Let $G = \mathbb Z/k_1 \times\cdots\times \mathbb Z/k_m$, a prodict of cyclic groups.  $X_2$ has $m$ 2-cells $e_1,\ldots, e_m$ and $m$ 3-cells $f_1,\ldots, f_m$, with $\partial f_i = k_i\cdot e_i$.

REPLY [6 votes]: This is a comment relating the other answers more than anything else.
Following are three isomorphic simplicial abelian groups which are Eilenberg-Maclane spaces $K(G,n)$.

The result of applying the Dold-Kan correspondence to the chain complex which has a copy of $G$ in degree $n$ and zero in all other degrees. This is an instance of the answer given by Tim Porter.
The result of taking the levelwise tensor product of $G$ with the free simplicial abelian group on the pointed simplicial set $\Delta^n/\partial\Delta^n$. This is an instance of the simplicial variant of the answer by BS.
The simplicial abelian group $\tilde{H}^{n-1}(sk^{n-1}\Delta^\bullet,G)$, where $\tilde{H}^\ast$ is reduced cohomology, $sk^k$ denotes the k-th skeleton, and $\Delta^\bullet$ stands for the standard functor from the simplicial category $\Delta$ into topological spaces.

You can prove that these are all isomorphic by showing that the normalized chain complex associated to each of them is the chain complex concentrated in degree $n$ with $G$ at that level. This is obvious for the first one, and easy for the second one. It is also not terribly difficult for the last one: it starts with observing that $sk^k \Delta^n$ is homotopy equivalent to a wedge of $\binom{n}{k+1}$ spheres of dimension $k$ (the number $\binom{n}{k+1}$ of spheres is also the number of monomorphisms of $\Delta^k$ into $\Delta^{n-1}$).
This also recovers the number of copies of $G$ appearing in Andre's answer: $\binom{n}{2}$ copies of $G$ for the $n$-simplices. This binomial coefficient is also easily seen to be the number of non-degenerate $n$-simplices in  $\Delta^2/\partial\Delta^2$ for $n>0$, so the same answer could be easily obtained from the second description above.