#72321169

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#151085
commutator	C₃₂	C₂×C₄×C₈
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72286481
prime radical	C₂×#2262	C₂×C₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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:

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

Polycyclic presentation:

2^11|a^2=h,b^2=degk,c^2=f,d^2=gik,e^2=gik,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bd,c^a=cj,c^b=cij,d^a=dg,d^b=dg,d^c=djk,e^a=egk,e^b=eg,e^c=ejk,f^a=fk,f^b=fj,f^d=fk,f^e=fk,g^a=gi,g^b=gi,g^c=gk,h^b=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321290, #72321722, #72322125, …
maximal normal subgroup of	#72321290, #72321722, #72322125, …

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