TITLE: Anomalous phenomena
QUESTION [6 upvotes]: What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern?
This is most interesting when the situation considered is very simple and basic and where the exceptional cases are not merely the lowest-numbered ones: for example, the outer automorphism of the symmetric group $S_{6}$ (which exists for no other $S_{n}$), and the existence of non-standard differentiable structures on $\mathbb{R}^{4}$ (but no other $\mathbb{R}^{n}$).

REPLY [11 votes]: The integral of $x^r$ is another power of $x$, for any value of $r$ except $r=-1$, when it's a natural logarithm. That still amazes me.

REPLY [5 votes]: Besides the extra automorphism of $S_6$,
both $S_6$ and $S_7$ have exceptional triple covers.

REPLY [5 votes]: The special orthogonal group $\operatorname{SO}_8$ is the only
$\operatorname{SO}_n$ that has an outer automorphism of order $3$.