#72322082

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	#72321050
commutator	C₂²×C₃₂	C₂²×C₄
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#13742	C₄
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #11399 ◁ #149496 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

577 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:

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

Polycyclic presentation:

2^11|b^2=f,d^2=gi,e^2=gij,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bhj,c^a=cd,c^b=cdegjk,d^a=dg,d^b=dj,d^c=dg,e^a=egjk,e^b=ejk,e^c=egjk,f^a=fk,f^c=fjk,f^d=fk,f^e=fk,g^a=gi,g^b=gk,g^c=gi,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322125, #72323081, #72323115
maximal normal subgroup of	#72322125, #72323081, #72323115

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