Small order examples of non-nilpotent finite groups in which every minimal normal subgroup intersects the center nontrivially

Question

I know that $p$-groups have the property, and all nilpotent groups also do. I wanted to have some non-nilpotent examples. Then I checked quasisimple groups, and found every proper normal subgroups of a quasisimple is contained in the center.

Quasisimple group ${\rm SL}(2,q)$ would be an example I want, when $q>3$ is an odd. For an odd $q>3$, the center of ${\rm SL}(2,q)$ is of order 2 and that makes ${\rm SL}(2,q)$ a non-simple quasisimple group with the property held; otherwise, for an even $q>3$, ${\rm SL}(2,q)$ has trivial center. Although ${\rm SL}(2,3)$ is not quasisimple, ${\rm SL}(2,3)$ also holds the property and is hopefully the example of the smallest order. My question is: How to prove ${\rm SL}(2,3)$ is the example of the smallest order? What are other kinds of groups that may be examples?

Thank you, any help will be appreciated!

Answer 1

Here's a quick way to check that $\mathrm{SL}_2(3)$ has the desired property using the computer algebra system GAP:

grps:=AllSmallGroups([1..24],IsNilpotentGroup,false);; 
list:=Filtered(grps,x->Order(Center(x))>1);  

for g in list do
Print(StructureDescription(g),"  ",IsSubgroup(Center(g),Socle(g)),"\n");
od;

C3 : C4  false
D12  false
C3 x S3  false
C5 : C4  false
D20  false
C3 : C8  false
SL(2,3)  true
C3 : Q8  false
C4 x S3  false
D24  false
C2 x (C3 : C4)  false
(C6 x C2) : C2  false
C2 x A4  false
C2 x C2 x S3  false