First, let me include the same disclaimer that goes in the first line of any article I write: all groups considered herein are finite.

Academically, I work with connecting the arithmetic structure of the set of character degrees $\bigl\{\ \chi(1)\ \big|\ \chi \in \operatorname{Irr}_{\mathbb C}(G)\ \bigr\}$ with the underlying group structure of a group $G$. In this setting, nonabelian $p$-groups are (generally speaking) a triviality which are quickly dispensed in favor of getting to more interesting considerations.

Which is to say, my general understanding of nilpotent groups is not where I want it to be. I certainly know the foundational, easier results, most of which are true across all primes (except possibly $p=2$ and occasionally $p=3$). The general theme of this discussion is meant to be: what is different/peculiar about the prime $p$ that makes $p$-groups different from $q$-groups over primes $q\ne p$, and when/in what ways/etc. does this difference manifest? With that, the specific question is:

What are ways that $p$-groups and $q$-groups behave differently for distinct primes $p$ and $q$?

This is very much intended to be a Big List question and will be requested into a community wiki accordingly.

As examples of what is being requested, consider the following context and subsequent questions. Independent of the specific value of $p$ and up to isomorphism,

there is exactly one group of order $p$,

there are exactly two groups of order $p^2$,

there are exactly five groups of order $p^3$ (three abelian and two extraspecial).

If it exists, what is a value $k$ for which there are different numbers of (isomorphism classes of) groups of order $p^k$ and $q^k$?

Is the smallest such $k$ known?

If the smallest $k$ is known, are the smallest primes $p$, $q$ satisfying the disparity known?

For a $p,q$ pair from (1), labelled so that there are more groups with order $p^k$ than $q^k$, does the same hold for

*all*$\ell\geq k$, that there are more groups with order $p^\ell$ than $q^\ell$? I would think so, just by taking direct products, but it is conceivable that there is some hiccup that may cause $q$ to have a huge skip from $q^{\ell-1}$ to $q^\ell$ that $p$ did not enjoy, and THAT would be VERY interesting.

As an example of a difference between primes, a peculiarity to the prime $2$ is that a $2$-group whose exponent is $2$ is necessarily abelian while there are nonabelian $p$-groups of exponent $p$ when $p$ is odd, namely extraspecial $p$-groups. In my undergrad days, I set a task for seeing if "$2$" could be generalized to "prime" and quickly discovered how the proof broke down, but did not discover extraspecial $p$-groups, the basic counterexamples, until grad school. With that as backdrop, I am particularly interested in examples highlighting the adage that "2 is the oddest prime."

This is intentionally very flexible, so have fun with it!