Besides the Poincare group the other five generators are:
-Dilations: these are the most obvious, they uniformly re-scale the coordinates. Basically scale transformations
-Special Conformal Transformations: these are less obvious, they generate translations of the inverted coordinates, so
$X^\mu$/$X^2$ -----> $X^\mu$/$X^2$ + $A^\mu$
@JamalS shows a figure with the geometrical interpretation of these inverted translations
See also for instance the math properties of the conformal group at
http://bolvan.ph.utexas.edu/~vadim/classes/13f/SCA.pdf besides the more basic wiki article
The conformal transformations are important for a number of reasons, among them
1) in 4D Lorentzian spacetime that symmetry group is a representation of SO(4,2),
From an answer in PSE at, this I part of the reason there is an AdS-CFT correspondence:
From Commutation relations of the generators of the conformal group:
"One very interesting thing about all this: you might ask, what is a spacetime where SO(4,2) really is just generalized rotations (as opposed to rotations + SCTs + dilatations)? Well, $AdS_5$ is one! This might be your first clue towards the existence of the AdS-CFT correspondence! A CFT in 3+1-dimensional spacetime obeys the same algebra as the isometries of $AdS_5$. See "ANTI-DE SITTER SPACE" by Ingemar Bengtsson - pages 1-5 give a nice concise introduction to AdS spacetime and it's isometries."
2)null cones in Minkowski spacetime transform to null cones under a conformal transformation. And these symmetries exist at null infinity in the horizon of a black hole, which Hawking and his colleagues used to conclude that there are other conserved hair in black holes, specifically soft hair, and that those may (not proven, but hinted at) carry the previously missing information that was understood to be lost as particles fall into the blac holes (or at least into the horizon).
See the paper at https://arxiv.org/abs/1601.00921
3) the whole area of conformal field theory.