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/usr/share/gromacs/top/random.dat is in gromacs-data 5.0.2-1.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

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GLY  H   8.39
GLY 1HA  3.97
GLY 2HA  3.97
GLY  HA1 3.97
GLY  HA2 3.97
ALA  H   8.25
ALA  HA  4.35
ALA 1HB  1.40
ALA  HB1 1.40
VAL  H   8.44
VAL  HA  4.18
VAL  HB  2.13
VAL 1HG1 0.96
VAL 1HG2 0.96
VAL HG11 0.96
VAL HG21 0.96
ILE  H   8.20
ILE  HA  4.22
ILE  HB  1.89
ILE 1HG2 0.94
ILE HG21 0.94
ILE 1HG1 1.30
ILE HG11 1.30
ILE 1HD1 0.89
ILE HD11 0.89
LEU  H   8.42
LEU  HA  4.39
LEU 1HB  1.65
LEU 2HB  1.65
LEU  HB1 1.65
LEU  HB2 1.65
LEU  HG  1.65
LEU 1HD1 0.92
LEU HD11 0.92
LEU 1HD2 0.92
LEU HD21 0.92
SER  H   8.38
SER  HA  4.50
SER 1HB  3.89
SER 2HB  3.89
SER  HB1 3.89
SER  HB2 3.89
THR  H   8.24
THR  HA  4.35
THR  HB  4.22
THR 1HG2 1.23
THR HG21 1.23
ASP  H   8.41
ASP  HA  4.77
ASP 1HB  2.80
ASP 2HB  2.80
ASP  HB1 2.80
ASP  HB2 2.80
GLU  H   8.37
GLU  HA  4.30
GLU 1HB  2.03
GLU 2HB  2.03
GLU  HB1 2.03
GLU  HB2 2.03
GLU 1HG  2.30
GLU 2HG  2.30
GLU  HG1 2.30
GLU  HG2 2.30
LYS  H   8.41
LYS  HA  4.36
LYS 1HB  1.81
LYS 2HB  1.81
LYS  HB1 1.81
LYS  HB2 1.81
LYS 1HG  1.47
LYS 2HG  1.47
LYS  HG1 1.47
LYS  HG2 1.47
LYS 1HD  1.71
LYS 2HD  1.71
LYS  HD1 1.71
LYS  HD2 1.71
LYS 1HE  3.02
LYS 2HE  3.02
LYS  HE1 3.02
LYS  HE2 3.02
ARG  H   8.27
ARG  HA  4.40
ARG 1HB  1.85
ARG 2HB  1.85
ARG  HB1 1.85
ARG  HB2 1.85
ARG 1HG  1.72
ARG 2HG  1.72
ARG  HG1 1.72
ARG  HG2 1.72
ARG 1HD  3.31
ARG 2HD  3.31
ARG  HD1 3.31
ARG  HD2 3.31
ASN  H   8.75
ASN  HA  4.76
ASN 1HB  2.80
ASN 2HB  2.80
ASN  HB1 2.80
ASN  HB2 2.80
GLN  H   8.41
GLN  HA  4.37
GLN 1HB  2.08
GLN 2HB  2.08
GLN  HB1 2.08
GLN  HB2 2.08
GLN 1HG  2.38
GLN 2HG  2.38
GLN  HG1 2.38
GLN  HG2 2.38
MET  H   8.42
MET  HA  4.51
MET 1HB  2.08
MET 2HB  2.08
MET  HB1 2.08
MET  HB2 2.08
MET 1HG  2.63
MET 2HG  2.63
MET  HG1 2.63
MET  HG2 2.63
MET 1HE  2.13
MET  HE1 2.13
CYS  H   8.31
CYS  HA  4.69
CYS 1HB  3.16
CYS 2HB  3.16
CYS  HB1 3.16
CYS  HB2 3.16
TRP  H   8.09
TRP  HA  4.70
TRP 1HB  3.26
TRP 2HB  3.26
TRP  HB1 3.26
TRP  HB2 3.26
TRP  HE1 9.99 
TRP  HD1 7.24
TRP  HE3 7.65
TRP  HZ3 7.17
TRP  HH2 7.24
TRP  HZ2 7.50
PHE  H   8.23
PHE  HA  4.66
PHE 1HB  3.11
PHE 2HB  3.11
PHE  HB1 3.11
PHE  HB2 3.11
PHE  HD1 7.34
PHE  HE1 7.34
PHE  HZ  7.34
TYR  H   8.18
TYR  HA  4.60
TYR 1HB  3.02
TYR 2HB  3.02
TYR  HB1 3.02
TYR  HB2 3.02
TYR  HD1 7.15
TYR  HE1 6.86
HIS  H   8.42
HIS  HA  4.63
HIS 1HB  3.23
HIS 2HB  3.23
HIS  HB1 3.23
HIS  HB2 3.23
HIS  HE1 8.12
HIS  HD1 7.14
PRO  HA  4.47
PRO 1HB  2.16
PRO 2HB  2.16
PRO  HB1 2.16
PRO  HB2 2.16
PRO 1HG  2.03
PRO 2HG  2.03
PRO  HG1 2.03
PRO  HG2 2.03
PRO 1HD  3.65
PRO 2HD  3.65
PRO  HD1 3.65
PRO  HD2 3.65