#72322785

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72322783
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂²×D₃₂	C₂×C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #11399 ◁ #149496 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

499 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bijk,c^a=cd,c^b=chij,d^b=dj,d^c=dg,e^b=ejk,e^c=eg,f^a=fj,f^c=fjk,f^d=fk,f^e=fk,g^b=gk,g^c=gi,h^a=hk,h^c=hk,i^c=ij,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322837, #72323081, #72323131
maximal normal subgroup of	#72322837, #72323081, #72323131

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