#72323081

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72323079
commutator	C₂²×C₃₂	C₂³×C₄
Frattini, cosocle	C₂×#1397	C₂⁴
prime radical	#72286862	C₄
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ #479914 ◁ #72231862 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

2451 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:

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

Polycyclic presentation:

2^12|b^2=g,f^2=hj,g^2=i,h^2=jk,j^2=kl,k^2=l,b^a=bk,c^b=ci,d^a=df,d^b=djl,d^c=de,e^b=el,f^a=fh,f^b=fkl,f^d=fh,g^a=gl,g^d=gk,g^f=gl,h^a=hj,h^b=hl,h^d=hj,i^d=il,j^a=jk,j^d=jk,k^a=kl,k^d=kl

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323149
maximal normal subgroup of	#72323149

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 4, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);