#72321260

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	C₄	#150160
commutator	C₂²×C₃₂	C₂²×C₄
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72321075
prime radical	#512934	C₄
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ C₂²×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

705 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:

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

Polycyclic presentation:

2^11|a^2=f,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bhk,c^a=ce,c^b=cd,d^a=dk,e^a=eg,e^c=egjk,f^c=fjk,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

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321290, #72323081, #72323095
maximal normal subgroup of	#72321290, #72323081, #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, 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.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*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, 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);