41063625 is the smallest such cube, from

41063625   56623104 66430125

56623104   66430125 41063625

66430125   41063625 56623104

The three numbers are the cubes of 345, 384 and 405.

These are the first three rows of output from

DefDbl A-Z

Dim crlf$, hist(90)

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 For n = 3 To 9999

   DoEvents

   cubect = 0

   cube = n * n * n

   s$ = LTrim(Str(cube)): h$ = s

   Do

     permute s

     If s = h Then Exit Do

     v = Val(s)

     cr = Int(v ^ (1 / 3) + 0.5)

     If cr * cr * cr = v And Left(s, 1) <> "0" Then cubect = cubect + 1: hist(cubect) = v

   Loop

   If cubect > 1 Then

    Text1.Text = Text1.Text & h & "  "

    For i = 1 To cubect

      Text1.Text = Text1.Text & Str(hist(i))

    Next

    Text1.Text = Text1.Text & crlf

   End If

 Next

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub

The program finds further values:

Filtering out duplicates (as each set is listed for each of its members):

1003003001   1030301000 1331000000

1006012008   1061208000 8012006001 8120601000

1086373952   1375036928 5097328361

1287913472   1897413272 3877292411

2062933417   3029741623 9274236301

3595640768   6740558369 7066834559

5204699837   6804992375 8402569937

The program was stopped before terminating. The list includes 1006012008 = 1002^3, which has 3 permutations that are cubes and are in fact cubes of permutations of 1002.