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2 typos corrected; deleted 1 characters in body

re 1): André's answer is superb, but just for the record, original references determining the homotopy groups of E8 are here: http://ncatlab.org/nlab/show/E8#HomotopyGroupsReferences

re 2):

One way to think about the phenomenon $B E_8 \simeq_{15} B^3 U(1) \simeq K(\mathbb{Z},4)$ from the point of view of string theory is to compare it to

a) the equivalence $B PU(\mathcal{H}) \mathrm{PU}(\mathcal{H}) \simeq B^2 U(1) \simeq K(\mathbb{Z},3)$ that controls Freed-Witten anomaly cancellation over D-branes

b) generalized complex geometry and exceptional generalized geometry that controls various other geometric structures in string theory.

In all these cases, one is looking at geometry which arises from reduction of structure groups along maps $G \to K$ of groups with the property that they are weak homotopy equivalences.

This is true for the inclusions of maximal compact subgroups that control generalized complex and exceptional generalized geometry, hence the U-duality symmetry of supergravity theories in various dimensions.

Notice that these inclusions are far from being equivalences as morphisms of Lie groups. But they are equivalences of the underlying topological spaces.

This situation now has a good analog in higher smooth geometry, which "explains" the role of $E_8$.

Namely, there is a smooth 3-group $\mathbf{B}^2 U(1)$ (a smooth group 2-stack) and the universal degree 4-class on $B E_8$ has a smooth refinement to a morphism of smooth 3-groups (group 2-stacks)

$$\Omega \mathbf{a} : E_8 \to \mathbf{B}^2 U(1) .$$

There is a higher analog of the notion of "reduction of structure groups" along such higher maps, and this controls the geometry of the supergravity C-field. For comparison, there is similarly a morphism of smooth 2-groups (smooth group stacks)

$$\Omega \mathbf{dd} : PU(\mathcal{H}) \mathrm{PU}(\mathcal{H}) \to \mathbf{B}U(1)$$

and its induced "generalized geometry" by "reduction of higher structure groups" controls precisely the Chan-Paton bundles on D-branres D-branes twisted by the $B$-field.

Both of these morphisms of smoth higher stacks becomes become equivalences of topological spaces under geometric realization (the first on 15-coskeleta, hence over the relevant spacetimes). So we may think of this as saying that, in a precise sense as saying that:

"The Lie group $E_8$ is a generalized maximal compact subgroup of the smooth 3-group $\mathbf{B}^2 U(1)$. The geometry of the $C$-field is the 'generalized geometry' controled by this 'inclusion'."

For more details on all this, see around section 4.3 of

http://arxiv.org/abs/1201.5277

and the big overview tables in section 4.4.1 of

http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf

1

re 1): André's answer is superb, but just for the record, original references determining the homotopy groups of E8 are here: http://ncatlab.org/nlab/show/E8#HomotopyGroupsReferences

re 2):

One way to think about the phenomenon $B E_8 \simeq_{15} B^3 U(1) \simeq K(\mathbb{Z},4)$ from the point of view of string theory is to compare it to

a) the equivalence $B PU(\mathcal{H}) \simeq B^2 U(1) \simeq K(\mathbb{Z},3)$ that controls Freed-Witten anomaly cancellation over D-branes

b) generalized complex geometry and exceptional generalized geometry that controls various other geometric structures in string theory.

In all these cases, one is looking at geometry which arises from reduction of structure groups along maps $G \to K$ of groups with the property that they are weak homotopy equivalences.

This is true for the inclusions of maximal compact subgroups that control generalized complex and exceptional generalized geometry, hence the U-duality symmetry of supergravity theories in various dimensions.

Notice that these inclusions are far from being equivalences as morphisms of Lie groups. But they are equivalences of the underlying topological spaces.

This situation now has a good analog in higher smooth geometry, which "explains" the role of $E_8$.

Namely, there is a smooth 3-group $\mathbf{B}^2 U(1)$ (a smooth group 2-stack) and the universal degree 4-class on $B E_8$ has a smooth refinement to a morphism of smooth 3-groups (group 2-stacks)

$$\Omega \mathbf{a} : E_8 \to \mathbf{B}^2 U(1) .$$

There is a higher analog of the notion of "reduction of structure groups" along such higher maps, and this controls the geometry of the supergravity C-field. For comparison, there is similarly a morphism of smooth 2-groups (smooth group stacks)

$$\Omega \mathbf{dd} : PU(\mathcal{H}) \to \mathbf{B}U(1)$$

and its induced "generalized geometry" by "reduction of higher structure groups" controls precisely the Chan-Paton bundles on D-branres twisted by the $B$-field.

Both of these morphisms of smoth higher stacks becomes equivalences of topological spaces under geometric realization (the first on 15-coskeleta, hence over the relevant spacetimes). So we may think of this as saying that, in a precise sense as saying that

"The Lie group $E_8$ is a generalized maximal compact subgroup of the smooth 3-group $\mathbf{B}^2 U(1)$. The geometry of the $C$-field is the 'generalized geometry' controled by this 'inclusion'."

For more details on all this, see around section 4.3 of

http://arxiv.org/abs/1201.5277

and the big overview tables in section 4.4.1 of

http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf