Discussion: I was curious too, so I checked. Forgive my brute force method.

Rules: Place 5 stars in a 5x5 grid. No stars can be in the same row, column, nor adjacent to each other.

I started with a star in the left most column, C1 and went column by column after that. The numbers represent the order in which I placed the stars. If the numbers stop, that means there's no more branches after that number. Xs indicate there's no valid solution to this setup.


~~~
Group1 (1st row group) (2 possibilities):

1)
1----
-----
-2---
-----
--3--

2)
1----
--3--
-----
-2---
-----

X
1----
--3--
-----
-----
-2---

X
1----
-----
--3--
-----
-2---


G2 (2nd row group) (3 possibilities):

1)
--3--
1----
---4-
-2---
-----

2)
--3--
1----
-----
-2---
---4-

X
--3--
1----
---4-
-----
-2---

X
--3--
1----
-----
---4-
-2---

3)
-----
1----
--3--
-----
-2---


G3 (4 possibilities) (Note that half of these are mirrors over the middle row):

1)
-2---
-----
1----
--3--
-----

2)
-2---
-----
1----
-----
--3--

3)
-----
--3--
1----
-----
-2---

4)
--3--
-----
1----
-----
-2---

G4 (3 possibilities): G4 group is the mirror of G2 over the middle row.

G5 (2 possibilities): G5 is the mirror of G1 over the middle row.

Total: 14 possibilities
~~~
> Translations

If we start at "1" then move to "2" then "3" and so on, of the 14 possibilities, I see 6 translation groups:
T1 - D2R1 (Down 2, Right 1).
G1#1, G2#1, G3#3, G4#3 and G5#2

T2 - D3R1 (Down 3, Right 1)
G1#2, G2#3, G3#1, G4#1, and G5#1

T3 - (D2R1x2 + D4R1 + D3R1)
G2#2

T4 - (D3R1x2 + D1R1 + D2R1)
G4#2

T5 - (D3R1+D4R1+D2R1x2)
G3#2

T6 - (D2R1+D1R1+D3R1x2)
G3#4

> Mirrors

The mirror (over the middle row) of D2R1 is D3R1. Mirror of D3R1 is D4R1. This means T1 and T2 are mirrors, T3 and T4 are mirrors, and T5 and T6 are mirrors. Thus we have 3:
```
M1 - The five translations of G1#1 times 2 for its mirror, gives 10 possibilities from this group.
1----
---4-
-2---
----5
--3--

M2 -  The one instance of this translation (G2#2) and its mirror gives 2 possibilites for this group.
--3--
1----
----5
-2---
---4-

M3 - The one instance of this translation (G3#2) and its mirror gives 2 possibilites for this group.
-2---
---4-
1----
----5
--3--
```
> Rotations

If we include 180 degree rotations around the center, M2 and M3 become related (G2#2 is G3#2 rotated), giving us 2 only groups: Rot1 (from M1) and Rot2 (M2 + M3)


If anyone has a big brain or non-brute force way of solving this, I'd be very happy to hear it!