#72322796

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	#72322794
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

684 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=defghik,d^2=gi,e^2=ghjk,f^2=hik,g^2=ij,h^2=j,i^2=jk,j^2=k,b^a=bk,c^a=cd,c^b=chi,d^a=djk,d^b=dk,d^c=dg,e^c=egj,e^d=ek,f^a=fj,f^c=fj,f^d=fk,f^e=fk,g^a=gk,g^c=gi,h^a=hk,h^c=hj,i^c=ij,j^c=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322837, #72323074, #72323095
maximal normal subgroup of	#72322837, #72323074, #72323095

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