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When you say lead I'm assuming you mean Graphite (pencil lead). That being said, I would guess (I stress I'm guessing) that Graphite has a higher heat capacity because it's delocalized bonds in the carbon rings allow it to absorb more heat than Diamond's single bonds.
I agree with Armin I think you might have mistaken lead for graphite. Diamond is known to be one of the best thermal conductors known, it is often a better thermal conductor than many metals including copper and aluminum.
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