Non solvable group requiring more than 2 generators The smallest group requiring more than 2 generators is
$$
C_2 \times C_2 \times C_2
$$
but that group is abelian. The smallest non abelian groups requiring more than 2 generators are groups with quotient $ C_2^3 $, such as $ D_8 \times C_2 $ and $ Q_8 \times C_2 $. But these extensions of $ C_2^3 $ are still solvable.
What is the smallest finite group $ G $ which is not solvable and requires more than two generators?
This question is the same as
The smallest group with 3 generators
but with "non abelian" replaced by "non solvable"
Kenta S points out that
$$
C_2^3 \times A_5
$$
is order 480 and non solvable and requires at least 3 generators (since $ C_2^3 $ is a quotient). I have a feeling that we can do better  and $ C_2^2 \times A_5 $ is also minimal 3 generated (EDIT: my feeling about $ 2^2A_5 $ was wrong see the answer from ahulpke or for an explicit 2 generation of $2^2A_5$ see answer from Parcly Taxel)(Also note that ahulpke found another non-solvable group of size 480 which is minimal 3 generated, namely $2^2S_5$).
 A: An explicit search in GAP finds $C_2\times C_2\times C_2\times A_5$ and $C_2\times C_2\times S_5$ to be the two nonsolvable groups of smallest order that are not 2-generated:
gap> l:=AllSmallGroups(Size,[60..480],IsSolvableGroup,false);;
gap> f:=FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> sel:=Filtered([1..Length(l)],x->Length(GQuotients(f,l[x]:findall:=false))=0);
[ 50, 51 ]
gap> List(l{sel},StructureDescription);
[ "C2 x C2 x S5", "C2 x C2 x C2 x A5" ]

(both are not 2-generated because of the quotient $C_2^3$). A similar search finds the smallest perfect group (no abelian quotient) to be of order 15360, of the form $C_2^4\rtimes(C_2^4\rtimes A_5)$ (Note that this structure does not determine the group uniquely, and only one of such groups is not 2-generated.)
Peripherally related, it is well-known that $A_5^{19}$ is 2-generated, while $A_5^{20}$ is not.
A: In fact $2^2\rm A_5$ can also be $2$-generated by pairing the extra transpositions with the generators of $\rm A_5$:
G := Group([(1,2,3)(6,7),(1,2,3,4,5)(8,9)]);
Print(StructureDescription(G));
C2 x C2 x A5

So the answer is $2^3\rm A_5$.
