#72322907

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, socle	C₂	#72322863
commutator, Frattini, cosocle	C₂²×C₃₂	C₂⁴
prime radical	C₂×#13742	C₂²
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ #33474 ◁ #512785 ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1203 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:

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

Polycyclic presentation:

2^11|b^2=g,c^2=g,f^2=hi,h^2=ij,i^2=jk,j^2=k,c^a=ck,c^b=cg,d^a=df,d^b=de,d^c=dejk,e^b=ek,f^a=fh,f^c=fk,f^d=fh,g^d=gk,h^a=hi,h^d=hi,i^a=ij,i^d=ij,j^a=jk,j^d=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322999
maximal normal subgroup of	#72322999

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