#72320679

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#495215
commutator	C₂²×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
socle	C₂	#72320212
prime radical	#7625194	C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #33420 ◁ #479917 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1956 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)(3,16)(4,11)(5,32)(6,13)(7,12)(8,14)(9,15)(10,21)(17,42)(18,61)(19,46)(20,62)(22,60)(23,44)(24,47)(25,78)(26,49)(27,86)(28,51)(29,52)(30,50)(31,53)(33,54)(34,55)(35,93)(36,94)(37,57)(38,97)(39,56)(40,59)(41,92)(43,67)(45,71)(48,81)(58,64)(63,99)(65,100)(66,102)(68,121)(69,101)(70,104)(72,103)(73,118)(74,106)(75,115)(76,108)(77,107)(79,109)(80,110)(82,125)(83,112)(84,113)(85,114)(87,126)(88,116)(89,117)(90,124)(91,105)(95,111)(96,119)(98,120)(122,128)(123,127);(3,17,19,41,21,22,23,33)(4,24,28,31)(6,37)(10,42,44,92,16,60,46,54)(11,47,51,53)(13,57)(18,65,66,63,67,95,69,68)(20,38,70,96,71,64,72,98)(25,79,73,82,81,90,91,80)(26,35)(27,75,30,77)(29,89,88,76)(40,84)(43,100,101,99,61,111,102,121)(45,97,103,119,62,58,104,120)(48,109,105,125,78,124,118,110)(49,93)(50,107,86,115)(52,117,116,108)(59,113)(74,85)(83,123,87,122)(106,114)(112,127,126,128);(1,3)(2,10)(4,25)(5,33)(6,38)(7,41)(8,19)(9,21)(11,48)(12,54)(13,58)(14,44)(15,16)(17,36)(18,29)(20,26)(22,39)(23,34)(24,73)(27,79)(28,81)(30,90)(31,91)(32,92)(35,71)(37,64)(40,96)(42,56)(43,52)(45,49)(46,55)(47,105)(50,109)(51,78)(53,118)(57,97)(59,120)(60,94)(61,116)(62,93)(63,122)(65,83)(66,76)(67,88)(68,123)(69,89)(70,85)(72,74)(75,82)(77,80)(84,98)(86,124)(87,95)(99,127)(100,126)(101,108)(102,117)(103,114)(104,106)(107,125)(110,115)(111,112)(113,119)(121,128);(1,4,26,83,7,30,84,76,34,24,74,122,36,77,37,88,9,28,35,87,5,27,40,89,8,31,85,123,39,75,6,29)(2,11,49,112,12,50,113,108,55,47,106,128,94,107,57,116,15,51,93,126,32,86,59,117,14,53,114,127,56,115,13,52)(3,18,38,82,22,68,70,91,19,69,96,79,33,95,71,81,21,67,64,80,17,63,72,73,23,66,98,90,41,65,20,25)(10,43,58,110,42,99,103,118,44,102,120,124,92,100,62,78,16,61,97,125,60,121,104,105,46,101,119,109,54,111,45,48)

Polycyclic presentation:

2^11|b^2=f,d^2=gi,e^2=jk,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bh,c^a=ck,c^b=ce,d^b=djk,d^c=dg,e^a=ek,e^c=ej,f^c=fj,f^d=fk,g^b=gk,g^c=gi,h^c=hk,i^c=ij,j^c=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321025, #72323055, #72323115
maximal normal subgroup of	#72321025, #72323055, #72323115

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