The fractal dimensionality of the infinite cluster at the percolation threshold for dimensionalities d>6 is shown to be D=4 (rather than the naive finite size scaling prediction D=d-2). Similarly, the conductivity of a sample of size L scales as L-d (rather than L-6). This anomalous behaviour is related to a dangerous irrelevant variable, associated with the probability to have vertices of three bonds. The crossover to the 'homogeneous' behaviour occurs at length scales which are short compared with the correlation length. The 'links and blobs' picture is confirmed for d>6, and the size of the latter is estimated.