Is the Centre of a Group with 4 Conjugacy Classes of Order 20 Trivial?

[playa007]

Homework Statement

If a group has order 20 has 4 conjugacy classes, it must have a trivial centre. True or False?

Homework Equations

The Class Equation

The Attempt at a Solution

I believed the answer to be false with this following counterexample:
20 = lZ(G)l + (20/4 + 20/4 + 20/5 + 20/5) => order of Z(G) = 2 => the centre is non-trivial.
where 5, 5, 4, 4 is the size of the 4 conjugate classes respectively

Any feedback on my reasoning and any flaws are very much appreciated

 

[morphism]

You haven't demonstrated that such a group exists.