#72227053

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227052
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#512928	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,c^2=eijk,d^2=g,e^2=h,f^2=ij,g^2=k,h^2=k,i^2=jk,j^2=k,b^a=bf,c^a=ci,c^b=cfik,d^c=dg,e^a=ejk,e^c=ejk,f^a=fi,f^b=fijk,f^c=fijk,f^e=fk,g^c=gk,h^a=hk,h^c=hk,i^a=ij,i^b=ijk,i^c=ijk,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	#72227066, #72227158, #72227166
maximal normal subgroup of	#72227066, #72227158, #72227166

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