#72227068

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216590
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	C₂×#13692	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

936 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:

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

Polycyclic presentation:

2^11|a^2=d,d^2=g,e^2=hk,f^2=ijk,g^2=k,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cf,c^b=cehk,d^b=dj,d^c=di,e^a=eh,e^b=ehj,e^c=ehj,e^d=ek,f^a=fjk,f^b=fjk,f^c=fijk,g^b=gk,g^c=gk,h^a=hj,h^b=hjk,h^c=hjk,i^c=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227077, #72227138, #72227171
maximal normal subgroup of	#72227077, #72227138, #72227171

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, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.11);;
gap> SetPower(coll, 6, F.9*F.10*F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
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*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, 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);