Center of binary octahedral group

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) binary octahedral group (see subgroup structure of binary octahedral group).
The subgroup is a normal subgroup and the quotient group is isomorphic to symmetric group:S4.
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Definition

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Arithmetic functions

Function	Value	Explanation
order of the whole group	48
order of the subgroup	2
index of the subgroup	24	Follows from Lagrange's theorem.
size of conjugacy class = index of normalizer	1	center is normal
number of conjugacy classes in automorphism class	1	center is characteristic

Subgroup-defining functions

Subgroup-defining function	What it means in general	Why it takes this value
center	set of elements that commute with every group element
third derived subgroup	derived subgroup of derived subgroup of derived subgroup	derived subgroup is isomorphic to SL(2,3), second derived subgroup is isomorphic to quaternion group
socle	join of all minimal normal subgroups	It is the unique minimal normal subgroup. The group is a monolithic group.
Frattini subgroup	intersection of all maximal subgroups