Graham Ellis would be able to better comment on the correctness of his code for $SL(5,\mathbb Z)$, as he appears to be the author of the HAP package in GAP.

But his code executes quickly and claims to compute the homology of $SL(5,\mathbb Z)$. 

$$H_2(SL_5(\mathbb Z), \mathbb Z)$$

    LoadPackage("HAP");;
    R:=ResolutionArithmeticGroup("SL(5,Z)",4);;
    Homology(TensorWithIntegers(R),2);

Returns $[2, 12]$, which I believe means the code is claiming this homology $H_2$ group is isomorphic to $\mathbb Z_2 \oplus \mathbb Z_{12}$.  It similarly computes $H_3$ to be $\mathbb Z_2 \oplus \mathbb Z_{24}$, and $H_4$ to $\mathbb Z_{12}$. 

I believe the 4 in the call for the resolution is a reference to the length of the resolution, so 4 is longer than required for $H_2$.  With the resolution of length 4 call, the code chokes on an $H_5$ computation.