#72321215

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#513082
commutator, Frattini, cosocle	C₂²×C₃₂	C₂⁴
socle	C₂	#72321065
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ C₂²×#1397 ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1413 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:

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

Polycyclic presentation:

2^11|d^2=g,f^2=hi,h^2=ij,i^2=jk,j^2=k,b^a=bg,c^a=cf,c^b=ce,d^a=dk,d^c=dj,e^a=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	#72321290, #72322999, #72323128
maximal normal subgroup of	#72321290, #72322999, #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, 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, 2, 1, F.2*F.7);;
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, 1, 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);