#72322666

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72322665
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#41	#357
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

632 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:

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

Polycyclic presentation:

2^11|a^2=f,c^2=d,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,c^a=ce,c^b=chk,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	#72322689, #72323095, #72323123
maximal normal subgroup of	#72322689, #72323095, #72323123

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, 3, F.4);;
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.8*F.11);;
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);