#72322120

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	C₂×#2262	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

464 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)(3,18,22,77,24,78,47,80)(4,26,31,96)(5,34,38,97)(6,37)(7,44)(8,46)(11,52,48,54,57,87,74,122)(12,58,61,102)(13,63,66,67)(14,60)(16,17)(19,81,72,55,83,82,119,125)(20,42)(21,84,89,25,91,45,79,95)(23,92,76,90,94,36,114,111)(27,99,49,62)(28,103,105,118,106,75,107,41)(29,30)(32,110,85,39)(33,88,100,98)(35,112,68,116)(40,108)(43,86)(50,65)(51,73)(56,113,117,64,126,104,124,128)(59,93,101,123,127,70,120,121)(69,109)(71,115);(1,2,8,16,9,17,46,10)(3,19)(4,27,100,12,31,49,33,61)(5,35,85,66,38,68,32,13)(6,40,30,50,43,14,69,71)(7,15,44,73,20,53,42,51)(11,45)(18,55,78,125)(21,87,91,52)(22,72)(23,93)(24,83)(25,74)(26,58,88,99,96,102,98,62)(28,104,106,113)(29,108,37,115,109,60,86,65)(34,67,110,112,97,63,39,116)(36,59,92,127)(41,117)(47,119)(48,95)(54,89,122,79)(56,75)(57,84)(64,105,128,107)(70,94)(76,123)(77,82,80,81)(90,120,111,101)(103,126)(114,121)(118,124);(1,3,20,84,46,22,44,25,9,24,7,45,8,47,42,95)(2,11,53,119,10,48,73,19,17,57,15,72,16,74,51,83)(4,28,32,111,100,107,5,36,31,106,85,90,33,105,38,92)(6,41,86,94,69,103,109,114,43,118,37,23,30,75,29,76)(12,59,35,113,49,120,66,128,61,127,68,104,27,101,13,64)(14,70,115,126,50,121,108,124,40,93,65,56,71,123,60,117)(18,34,91,98,80,110,89,26,78,97,21,88,77,39,79,96)(52,67,81,62,122,112,125,58,87,63,82,99,54,116,55,102)

Polycyclic presentation:

2^11|a^2=f,b^2=fhj,c^2=fik,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=dg,e^a=egjk,e^b=ejk,e^c=egjk,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, #72323074, #72323128
maximal normal subgroup of	#72322125, #72323074, #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, 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);;
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);;
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, 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);