#72322105

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72321083
commutator	C₂²×C₃₂	C₂²×C₄
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#41	#367
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

472 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)(3,16,19,66,21,67,37,69)(4,23,26,80)(5,29)(6,34)(7,36)(10,41,43,105,17,42,63,107)(11,46,12,50)(13,56)(14,58)(18,31)(20,76,75,126,77,128,99,93)(22,79,35,68,78,86,72,127)(24,84,85,98,87,65,88,30)(25,32)(27,92,73,89)(28,95,91,83)(33,74)(38,59)(39,62)(44,111,110,123,70,116,120,125)(45,112,61,106,71,124,100,109)(47,104,115,57,53,103,117,97)(48,101)(49,60)(51,118,54,119)(52,122,55,114)(64,94,82,81)(90,96)(102,113,108,121);(1,2,8,9)(3,17)(4,12,26,11)(5,13,33,38)(6,14,18,39)(7,15,36,40)(10,21)(16,42)(19,63)(20,70)(22,71)(23,46,80,50)(24,47)(25,48,90,49)(27,54,73,51)(28,55,91,52)(29,59,74,56)(30,97)(31,58,34,62)(32,60,96,101)(35,100)(37,43)(41,67)(44,77)(45,78)(53,87)(57,98)(61,72)(64,102,82,108)(65,103)(66,107)(68,109)(69,105)(75,120)(76,116)(79,124)(81,113,94,121)(83,114,95,122)(84,104)(85,115)(86,112)(88,117)(89,119,92,118)(93,123)(99,110)(106,127)(111,128)(125,126);(1,3,18,72,36,19,34,22,8,21,6,35,7,37,31,78)(2,10,39,100,40,43,62,45,9,17,14,61,15,63,58,71)(4,24,27,93,91,88,81,128,26,87,73,126,28,85,94,76)(5,30,74,77,96,84,90,99,33,98,29,20,32,65,25,75)(11,47,51,123,55,117,121,111,12,53,54,125,52,115,113,116)(13,57,56,70,101,103,49,120,38,97,59,44,60,104,48,110)(16,64,79,83,69,92,127,23,67,82,86,95,66,89,68,80)(41,102,112,114,107,118,109,46,42,108,124,122,105,119,106,50)

Polycyclic presentation:

2^11|a^2=f,b^2=hk,c^2=fik,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=dg,e^a=eg,e^c=eg,f^c=fjk,f^d=fk,f^e=fk,g^a=gi,g^c=gi,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, #72323105
maximal normal subgroup of	#72322125, #72323087, #72323105

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, 3, F.6*F.9*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);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
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);;
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);