  It is well-known that the expected scaled maximum of non-negative random
variables with unit mean defines a stable tail dependence function associated
with some extreme-value copula. In the special case when these random variables
are independent and identically distributed, min-stable multivariate
exponential random vectors with the associated survival extreme-value copulas
are shown to arise as finite-dimensional margins of an infinite exchangeable
sequence in the sense of De Finetti's Theorem. The associated latent factor is
a stochastic process which is strongly infinitely divisible with respect to
time, which induces a bijection from the set of distribution functions F of
non-negative random variables with finite mean to the set of L\'evy measures on
the positive half-axis. Since the Gumbel and the Galambos copula are the most
popular examples of this construction, the investigation of this bijection
contributes to a further understanding of their well-known analytical
similarities. Furthermore, a simulation algorithm based on the latent factor
representation is developed, if the support of F is bounded. Especially in
large dimensions, this algorithm is efficient because it makes use of the De
Finetti structure.
