#72303723

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10918
commutator	C₂×#41	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#124038
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

3823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,13,8,15,35,46)(3,16,20,51,22,47,36,39)(4,23,27,45,29,50,37,10)(5,11,30,12,31,14,6,32)(9,21,40,19,41,18,49,33)(17,54,56,53,58,42,59,64)(24,38,55,61,57,62,60,63)(25,44,34,52,28,48,26,43);(1,3)(2,9)(4,24,29,57)(5,18,31,21)(6,33,30,19)(7,36)(8,22)(10,42,45,54)(11,39,14,51)(12,47,32,16)(13,49)(15,41)(17,25)(20,35)(23,53,50,64)(26,56)(27,60,37,55)(28,58)(34,59)(38,43)(40,46)(44,63)(48,61)(52,62);(1,4)(2,10,15,45)(3,17)(5,25)(6,34)(7,37)(8,29)(9,38,41,62)(11,43,14,52)(12,48,32,44)(13,50,46,23)(16,53,47,64)(18,24)(19,55)(20,59)(21,57)(22,58)(26,30)(27,35)(28,31)(33,60)(36,56)(39,42,51,54)(40,63,49,61)

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=dghij,f^2=j,g^2=j,b^a=be,c^a=cegj,c^b=cd,d^a=di,d^b=dij,e^a=ej,e^b=ei,e^c=eij,e^d=ej,f^b=fj,f^c=fj,g^a=gj,g^b=gi,g^c=gij,g^d=gj,h^a=hi,h^c=hij,h^d=hj,h^e=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303732, #72312878, #72313024
maximal normal subgroup of	#72303732, #72312878, #72313024

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);