#72321714

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72286684
commutator	C₂×C₃₂	C₂²×C₈
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

495 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=fhj,c^2=fik,d^2=jk,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bj,c^a=ce,c^b=cd,d^a=dj,d^b=dk,d^c=dj,e^a=eg,e^b=ejk,e^c=eg,f^c=fjk,f^e=fk,g^a=gi,g^b=gk,g^c=gi,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321722, #72323123, #72323128
maximal normal subgroup of	#72321722, #72323123, #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.6);;
gap> SetPower(coll, 2, F.6*F.8*F.10);;
gap> SetPower(coll, 3, F.6*F.9*F.11);;
gap> SetPower(coll, 4, F.10*F.11);;
gap> SetPower(coll, 5, F.7*F.9*F.10*F.11);;
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.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*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, 3, F.6*F.10*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.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);