Requirements:

1. GAP small groups library
2. data file normalized_invariants.txt of values from Table 1
3. data file small_groups_in_isoclinism_families.txt of small group id's of groups in isoclinism families (families $\Gamma_i$ and $\Phi_i$ are merged for $i=2,\dots,5$, and $\Gamma_i$ and $\Phi_{i+1}$ are merged for $i=6,7,8$).

Code for computing $A_G(t)$ and $B_G(t)$

R = PolynomialRing(QQ,'p,t')
p,t = R.gens()
FR = FractionField(R)
def A(G, t = None):
    if t is None:
        R = FractionField(PolynomialRing(ZZ, 't'))
        t = R.gen()
    return sum([1/(1-Integer(gap.Order(gap.Centralizer(G,x)))*t) for x in gap.Elements(G)])/Integer(gap.Order(G))

def B(G, t = None):
    if t is None:
        R = FractionField(PolynomialRing(ZZ, 't'))
        t = R.gen()
    return (1 + sum([t*B(gap.Centralizer(G,gap.Representative(C)), t = t) for C in gap.ConjugacyClasses(G) if gap.Size(C) > 1]))/(1 - Integer(gap.Size(gap.Center(G)))*t)

SG = gap.SmallGroup

def ASG(n,k):
    return A(SG(n,k))

def BSG(n,k):
    return B(SG(n,k))

Load values from Table 1

AD = dict()
BD = dict()
f = file("normalized_invariants.txt", 'r')
for l in f:
    if l[0]!="#":
        el = eval(l)
        AD[el[0]] = el[1]
        BD[el[0]] = el[2]
f.close()

Load list of small groups in isoclinism families

For each isoclinism family of rank up to five, the GAP ID's of all groups of order up to $\min(p^5,100)$ are collected.

f = file('small_groups_in_isoclinism_families.txt', 'r')
idlist_family = dict()
for l in f:
    el = eval(l)
    idlist_family[el[0]] = el[1]

Code for verifying values in Table 1 against computation using GAP groups

for k in idlist_family.keys():
    print k
    for id in idlist_family[k]:
        g = id[0]
        pg = prime_divisors(g)[0]
        formal_A = AD[k].substitute(p=pg).substitute(t=t*g)
        formal_B = BD[k].substitute(p=pg).substitute(t=t*g)
        print id, formal_A == ASG(*id), formal_B == BSG(*id)

 phi10
(243, 28) True True
(243, 29) True True
(243, 30) True True
phi7
(32, 43) True True
(32, 44) True True
(243, 56) True True
(243, 57) True True
(243, 58) True True
(243, 59) True True
(243, 60) True True
phi6
(243, 3) True True
(243, 4) True True
(243, 5) True True
(243, 6) True True
(243, 7) True True
(243, 8) True True
(243, 9) True True
phi5
(32, 49) True True
(32, 50) True True
(243, 65) True True
(243, 66) True True
phi4
(32, 27) True True
(32, 28) True True
(32, 29) True True
(32, 30) True True
(32, 31) True True
(32, 32) True True
(32, 33) True True
(32, 34) True True
(32, 35) True True
(243, 37) True True
(243, 38) True True
(243, 39) True True
(243, 40) True True
(243, 41) True True
(243, 42) True True
(243, 43) True True
(243, 44) True True
(243, 45) True True
(243, 46) True True
(243, 47) True True
phi3
(16, 7) True True
(16, 8) True True
(16, 9) True True
(32, 9) True True
(32, 10) True True
(32, 11) True True
(32, 13) True True
(32, 14) True True
(32, 15) True True
(32, 39) True True
(32, 40) True True
(32, 41) True True
(32, 42) True True
(81, 7) True True
(81, 8) True True
(81, 9) True True
(81, 10) True True
(243, 13) True True
(243, 14) True True
(243, 15) True True
(243, 16) True True
(243, 17) True True
(243, 18) True True
(243, 19) True True
(243, 20) True True
(243, 51) True True
(243, 52) True True
(243, 53) True True
(243, 54) True True
(243, 55) True True
(625, 7) True True
(625, 8) True True
(625, 9) True True
(625, 10) True True
phi2
(8, 3) True True
(8, 4) True True
(16, 3) True True
(16, 4) True True
(16, 6) True True
(16, 11) True True
(16, 12) True True
(16, 13) True True
(32, 2) True True
(32, 4) True True
(32, 5) True True
(32, 12) True True
(32, 17) True True
(32, 22) True True
(32, 23) True True
(32, 24) True True
(32, 25) True True
(32, 26) True True
(32, 37) True True
(32, 38) True True
(32, 46) True True
(32, 47) True True
(32, 48) True True
(27, 3) True True
(27, 4) True True
(81, 3) True True
(81, 4) True True
(81, 6) True True
(81, 12) True True
(81, 13) True True
(81, 14) True True
(243, 2) True True
(243, 11) True True
(243, 12) True True
(243, 21) True True
(243, 24) True True
(243, 32) True True
(243, 33) True True
(243, 34) True True
(243, 35) True True
(243, 36) True True
(243, 49) True True
(243, 50) True True
(243, 62) True True
(243, 63) True True
(243, 64) True True
(125, 3) True True
(125, 4) True True
(625, 3) True True
(625, 4) True True
(625, 6) True True
(625, 12) True True
(625, 13) True True
(625, 14) True True
(343, 3) True True
(343, 4) True True
phi9
(32, 18) True True
(32, 19) True True
(32, 20) True True
(243, 25) True True
(243, 26) True True
(243, 27) True True
phi8
(32, 6) True True
(32, 7) True True
(32, 8) True True
(243, 22) True True