Update Nov 2, 2011: I was at a conference in September, where G. Mason pointed out to me that $H^4(M,\mathbb{Z})$ probably contains an element of order 8, and therefore also $\mathbb{Z}/24\mathbb{Z}$. I believe the argument was the following: there is an order 8 element $g$ whose centralizer in the monster acts projectively on the unique irreducible $g$-twisted module of the monster vertex algebra $V^\natural$, such that one needs to pass to a cyclic degree 8 central extension to get an honest action. Rather than just looking at $L_0$-eigenvalues, one needs to examine character tables to eliminate smaller central extensions here. Naturally, like the claims I described before, the validity of this argument depends on some standard conjectures about the structure of twisted modules.
There is some evidence from characters that $H^4(M,\mathbb{Z})$ contains $\mathbb{Z}/12\mathbb{Z}$. In particular, the conjugacy class 24J (made from certain elements of order 24) has a character of level 288, and the corresponding irreducible twisted modules have a character whose expansion is in powers of $q^{1/288}$. Fusion in a cyclic group generated by a 24J element then yields a $1/12$ discrepancy in $L_0$-eigenvalues (L_0$-eigenvalues, meaning you will pick up 12th roots of unity from the associator)associator. This suggests that pulling If you pull back along a pointed map$B(\mathbb{Z}/24\mathbb{Z}) \to BM$corresponding to an element in class 24J yields (i.e., if you forget about twisted modules outside this cyclic group) you get a cocycle of order 12. This is the largest order you can get by this method - everything else divides 12. I don't know how the cocycles corresponding to different cyclic groups fit together. I don't know if you've seen Mason's paper, Orbifold conformal field theory and cohomology of the monster, but it is about related stuff. I don't understand how he got his meta-theorem with the number 48 at the end, though. As far as implications or meaning of the cocycle, all I can say is that the automorphism 2-group of the category of twisted modules of$V^\natural$has the monster as its truncation, and its 2-group structure is nontrivial. I've heard some speculation about twisting monster-equivariant elliptic cohomology, but I don't understand it. If you believe in AdS/CFT, this might say something about pure quantum gravity in 3 dimensions, but I have no idea what that would be. 1 There is some evidence from characters that$H^4(M,\mathbb{Z})$contains$\mathbb{Z}/12\mathbb{Z}$. In particular, the conjugacy class 24J (made from certain elements of order 24) has a character of level 288, and the corresponding irreducible twisted modules have a character whose expansion is in powers of$q^{1/288}$. Fusion then yields a$1/12$discrepancy in$L_0$-eigenvalues (meaning you will pick up 12th roots of unity from the associator). This suggests that pulling back along a pointed map$B(\mathbb{Z}/24\mathbb{Z}) \to BM\$ corresponding to an element in class 24J yields a cocycle of order 12. I don't know how the cocycles corresponding to different cyclic groups fit together.