M16

Definition
The group, sometimes denoted $$M_{16}$$ or $$M_4(2)$$, is defined as follows:

$$M_{16} = \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^5 \rangle$$.

Here, $$e$$ denotes the identity element.

1-isomorphism
The group is 1-isomorphic to the group direct product of Z8 and Z2. In other words, there is a bijection between the groups that restricts to an isomorphism on all cyclic subgroups on either side. The 1-isomorphism is explained by the cocycle halving generalization of Baer correspondence, where the intermediary is a class two Lie cring.

Subgroups
To describe subgroups, we use the defining presentation given at the beginning:

$$M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$$.

Smallest of its kind

 * This is a minimum order example of a non-abelian finite group that is 1-isomorphic to an abelian group -- it is 1-isomorphic to direct product of Z8 and Z2. It is, however, not the only such example: the other example is central product of D8 and Z4. See element structure of groups of order 16 for more details.
 * This is a minimum order example of a nilpotent group that is not a UL-equivalent group, i.e., the upper central series and lower central series are not the same. However, it is not the only example. In fact, all groups of order 16 and class two share this property with it. The other examples are SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, direct product of D8 and Z2, direct product of Q8 and Z2, and central product of D8 and Z4.
 * This is a minimum order example of a situation where a group has two characteristic subgroups that are both isomorphic to each other but are distinct. Both these are cyclic subgroups of order four. It is, however, not the only example of this order; there are other examples where a similar behavior occurs, albeit for different orders of subgroups -- for instance, order 2 (in nontrivial semidirect product of Z4 and Z4).

Description by presentation
The group can be defined using a presentation as follows:

gap> F := FreeGroup(2);; gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-5)];  gap> IdGroup(G); [ 16, 6 ]