#72322790

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	C₂×#2262	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

483 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=f,c^2=defgk,d^2=gijk,e^2=gi,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bij,c^a=cd,c^b=chijk,d^b=dj,d^c=dg,e^b=ejk,e^c=egjk,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, #72323087, #72323128
maximal normal subgroup of	#72322837, #72323087, #72323128

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