#72321641

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72286704
commutator	C₂×C₃₂	C₂³×C₄
Frattini, cosocle	C₂²×C₃₂	C₂⁴
prime radical	#72321604	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 128 points:

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

Polycyclic presentation:

2^11|b^2=g,d^2=g,f^2=hi,h^2=ij,i^2=jk,j^2=k,c^a=cf,c^b=ce,d^a=dk,d^c=dj,e^b=ek,f^a=fh,f^c=fh,f^d=fk,g^c=gk,h^a=hi,h^c=hi,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, #72322999, #72323087
maximal normal subgroup of	#72321722, #72322999, #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.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
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);