#72230735

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#93877
commutator	C₃₂	C₂×C₃₂
Frattini, cosocle	#93666	C₂²
socle	C₂	#72230695
prime radical	C₂²	#93878
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #7329 ◁ #93666 ◁ G
LECS	G ▷ #93666 ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

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

Polycyclic presentation:

2^11|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bd,c^b=chjk,d^a=df,d^c=dj,e^b=ejk,e^d=ek,f^a=fh,f^c=fk,g^b=gk,h^a=hj,j^a=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230759, #72230761, #72230812
maximal normal subgroup of	#72230759, #72230761, #72230812

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