#72322086

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	#72321047
commutator	C₂²×C₃₂	C₂²×C₄
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂²×D₃₂	C₂×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

480 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)(19,84)(20,22)(21,91)(23,96)(24,80)(26,101)(27,104)(29,100)(30,99)(31,37)(32,98)(34,114)(35,116)(36,75)(38,113)(39,112)(41,83)(42,110)(44,68)(45,107)(46,109)(47,97)(48,82)(49,55)(50,63)(51,62)(52,73)(54,90)(56,86)(57,127)(58,85)(59,87)(61,92)(64,103)(65,118)(67,126)(69,70)(71,125)(72,108)(76,128)(78,124)(79,81)(88,106)(89,95)(93,117)(94,111)(102,122)(105,123)(115,119)(120,121);(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)(9,17,43,73,47,77,97,52)(11,54)(18,56,80,124)(20,87,89,53)(21,92)(22,49)(23,86)(24,58)(25,60,98,101,99,103,100,63)(27,105,106,115)(28,108,36,74,109,62,110,68)(33,69,112,114,113,66,37,118)(35,61,117,128)(41,78)(45,119)(48,59)(55,82,125,90)(57,88)(67,81,127,107)(71,95)(76,94)(79,123)(83,84,96,85)(91,102,111,120)(93,122)(104,126)(116,121);(1,3,9,48,40,23,97,95,8,24,47,54,6,41,43,22)(2,11,17,78,10,49,52,19,16,59,77,86,14,71,73,58)(4,27,39,91,29,107,5,35,30,106,31,111,32,81,38,117)(7,45,28,94,75,104,110,116,46,79,109,21,42,88,36,93)(12,61,34,115,50,120,70,127,64,128,65,105,26,102,13,67)(15,76,62,126,51,121,74,123,44,92,108,57,72,122,68,119)(18,37,89,99,83,33,82,100,80,112,20,25,96,113,90,98)(53,118,84,103,125,69,124,63,87,114,85,60,55,66,56,101)

Polycyclic presentation:

2^11|b^2=f,c^2=fhij,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=dgj,e^a=egjk,e^b=ejk,e^c=eg,f^a=fk,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, #72323081, #72323123
maximal normal subgroup of	#72322125, #72323081, #72323123

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, 3, F.6*F.8*F.9*F.10);;
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*F.10);;
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);;
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*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);