#72227111

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225151
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	C₂×#135	#106
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

718 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=dfi,c^2=djk,d^2=g,e^2=h,f^2=i,h^2=j,j^2=k,b^a=be,c^a=chj,c^b=cefhk,d^b=djk,d^c=dk,e^a=eh,e^b=eh,e^c=ehk,e^d=ek,f^a=fjk,f^b=fjk,f^c=fi,g^b=gk,h^a=hj,h^b=hj,h^c=hj,i^a=ik,i^b=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227121, #72227167, #72227168
maximal normal subgroup of	#72227121, #72227167, #72227168

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.6*F.9);;
gap> SetPower(coll, 3, F.4*F.10*F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);