#72321687

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72286748
commutator	C₂×C₃₂	C₂²×C₈
Frattini, cosocle	C₂×#1397	C₂³
prime radical	#512951	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=hk,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,c^a=ce,c^b=cd,d^b=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	#72321722, #72323095, #72323131
maximal normal subgroup of	#72321722, #72323095, #72323131

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.8*F.11);;
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, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, 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);