  We give a detailed description of the Veldkamp space of the smallest slim
dense near hexagon. This space is isomorphic to PG(7, 2) and its 2^8 - 1 = 255
Veldkamp points (that is, geometric hyperplanes of the near hexagon) fall into
five distinct classes, each of which is uniquely characterized by the number of
points/lines as well as by a sequence of the cardinalities of points of given
orders and/or that of (grid-)quads of given types. For each type we also give
its weight, stabilizer group within the full automorphism group of the near
hexagon and the total number of copies. The totality of (255 choose 2)/3 =
10795 Veldkamp lines split into 41 different types. We give a complete
classification of them in terms of the properties of their cores (i. e.,
subconfigurations of points and lines common to all the three hyperplanes
comprising a given Veldkamp line) and the types of the hyperplanes they are
composed of. These findings may lend themselves into important physical
applications, especially in view of recent emergence of a variety of closely
related finite geometrical concepts linking quantum information with black
holes.
