#72322097

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72321075
commutator	C₂²×C₃₂	C₂²×C₄
Frattini, cosocle	C₂×#1397	C₂³
prime radical	#512951	C₄
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

(2,9)(3,16,18,66,20,68,37,70)(4,22,25,81)(5,28)(6,33)(7,35)(10,41,43,105,17,42,63,108)(11,46,12,50)(13,56)(14,58)(19,74,73,88,77,98,89,127)(21,80,34,69,79,126,76,86)(23,29,85,65,87,78,67,99)(24,31)(26,94,92,90)(27,64,93,84)(30,36)(32,75)(38,59)(39,62)(44,113,112,124,71,123,118,117)(45,114,61,107,72,125,100,111)(47,97,106,103,53,57,110,104)(48,101)(49,60)(51,119,54,120)(52,102,55,109)(82,128,95,83)(91,96)(115,116,121,122);(1,2,8,9)(3,17)(4,12,25,11)(5,13,32,38)(6,14,36,39)(7,15,35,40)(10,20)(16,42)(18,63)(19,71)(21,72)(22,46,81,50)(23,47)(24,48,91,49)(26,54,92,51)(27,55,93,52)(28,59,75,56)(29,97)(30,58,33,62)(31,60,96,101)(34,100)(37,43)(41,68)(44,77)(45,79)(53,87)(57,78)(61,76)(64,102,84,109)(65,103)(66,108)(67,110)(69,111)(70,105)(73,118)(74,123)(80,125)(82,115,95,121)(83,116,128,122)(85,106)(86,107)(88,117)(89,112)(90,120,94,119)(98,113)(99,104)(114,126)(124,127);(1,3)(2,10)(4,23)(5,29)(6,34)(7,37)(8,20)(9,17)(11,47)(12,53)(13,57)(14,61)(15,63)(16,64)(18,35)(19,75)(21,33)(22,66)(24,89)(25,87)(26,74)(27,85)(28,77)(30,79)(31,99)(32,78)(36,76)(38,97)(39,100)(40,43)(41,102)(42,109)(44,56)(45,62)(46,105)(48,118)(49,112)(50,108)(51,123)(52,106)(54,113)(55,110)(58,72)(59,71)(60,103)(65,96)(67,93)(68,84)(69,83)(70,81)(73,91)(80,90)(82,127)(86,128)(88,95)(92,98)(94,126)(101,104)(107,116)(111,122)(114,120)(115,117)(119,125)(121,124)

Polycyclic presentation:

2^11|a^2=f,b^2=hk,d^2=gijk,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bhk,c^a=cd,c^b=cdegjk,d^a=dg,d^b=dk,d^c=dgjk,e^a=eg,e^c=egjk,f^c=fjk,f^d=fk,f^e=fk,g^a=gi,g^c=gik,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322125, #72323087, #72323095
maximal normal subgroup of	#72322125, #72323087, #72323095

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