#72321660

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72286707
commutator	C₂×C₃₂	C₂²×C₈
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#33982	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

1468 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:

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

Polycyclic presentation:

2^11|b^2=hjk,c^2=f,e^2=gi,f^2=h,g^2=ij,h^2=k,i^2=jk,j^2=k,c^a=ce,c^b=cd,d^b=dk,e^a=eg,e^c=egj,f^a=fj,f^e=fk,g^a=gi,g^c=gik,h^a=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, #72323034, #72323087
maximal normal subgroup of	#72321722, #72323034, #72323087

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.8*F.10*F.11);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, 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);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
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, 1, 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);