Muggles' Guide to Harry Potter/Characters/Abraxas Malfoy

Overview
Abraxas Malfoy was grandfather to Draco, and father to Lucius Malfoy. He knew Horace Slughorn and died of Dragon Pox, presumably at an old age.

Half-Blood Prince
In their first Potions class, Draco asks Slughorn if he remembers his grandfather. Slughorn replies that he does and is sorry to hear of his death but it was not unexpected, "Dragon Pox, at his age" before turning away.

Professor Slughorn mentions Draco's grandfather again in the Slug Club Christmas party.

Strengths
Though we don't know how exactly, Abraxas must have been a pretty notable wizard during his time, as implied by both Draco and Slughorn.

Relationships with Other Characters
Apparently distant friends with Horace Slughorn, who is possibly around his age; they may have attended Hogwarts at the same time. It is also possible that he was a student of Slughorn's.

Analysis
We know that Professor Slughorn is more than a bit of a social climber, and that he seems to cut off those people who he feels don't offer him an "in" into the higher-flying parts of society. The fact that he acknowledges knowing Abraxas Malfoy would indicate that there was some fame or accomplishment in Abraxas' past to attract Slughorn's attention. We never find out, though, exactly what that accomplishment is.

Abraxas is brought into the story, as far as we can see, only to cast a light on Slughorn and some specifics of his social climbing. In an earlier chapter, we see that Slughorn is willing to cultivate Marcus Belby as long as he has hopes of making a connection with Belby's still living famous uncle. As Abraxas is no longer alive and does not represent a useful connection. Slughorn is not particularly interested in those who had connections to him. We suspect that this particular shallowness in Slughorn's character is one of the things that Harry dislikes, and one of the reasons Harry avoids the Slug Club.