#72322113

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	#513315	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

561 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)(3,18,21,78,23,80,47,82)(4,25,30,98)(5,33,37,99)(6,36)(7,43)(8,45)(11,52,48,53,56,88,74,124)(12,57,60,104)(13,62,65,66)(14,59)(16,17)(19,83,71,54,86,84,119,126)(20,87,91,24,93,44,81,97)(22,94,77,92,96,115,109,35)(26,101,49,61)(27,40,106,76,107,90,79,111)(28,29)(31,89,112,38)(32,75,102,100)(34,113,67,117)(39,108)(41,46)(42,95)(50,64)(51,72)(55,125,120,114,127,63,118,105)(58,122,103,85,123,128,121,69)(68,110)(70,116);(1,2,8,16,9,17,45,10)(3,19)(4,26,102,12,30,49,32,60)(5,34,112,65,37,67,31,13)(6,39,29,50,42,14,68,70)(7,15,43,72,46,73,41,51)(11,44)(18,54,80,126)(20,88,93,52)(21,71)(22,69)(23,86)(24,74)(25,57,75,101,98,104,100,61)(27,105,107,114)(28,108,36,116,110,59,95,64)(33,66,89,113,99,62,38,117)(35,58,92,123)(40,118)(47,119)(48,97)(53,91,124,81)(55,76)(56,87)(63,106,125,79)(77,122)(78,84,82,83)(85,96)(90,120)(94,103,115,121)(109,128)(111,127);(1,3)(2,11)(4,27)(5,35)(6,40)(7,44)(8,47)(9,23)(10,48)(12,58)(13,63)(14,69)(15,71)(16,74)(17,56)(18,75)(19,72)(20,89)(21,45)(22,95)(24,43)(25,78)(26,103)(28,109)(29,111)(30,107)(31,94)(32,106)(33,91)(34,114)(36,96)(37,92)(38,93)(39,85)(41,97)(42,90)(46,87)(49,121)(50,122)(51,86)(52,101)(53,57)(54,62)(55,116)(59,118)(60,123)(61,88)(64,127)(65,125)(66,126)(67,105)(68,76)(70,128)(73,119)(77,110)(79,102)(80,100)(81,99)(82,98)(83,117)(84,113)(104,124)(108,120)(112,115)

Polycyclic presentation:

2^11|a^2=f,b^2=fhj,d^2=gijk,e^2=gi,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bhjk,c^a=cd,c^b=cdegk,d^a=dg,d^b=dj,d^c=dgjk,e^a=egjk,e^b=ejk,e^c=egk,f^c=fjk,f^d=fk,f^e=fk,g^a=gi,g^b=gk,g^c=gik,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, #72323055, #72323128
maximal normal subgroup of	#72322125, #72323055, #72323128

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