#72267889

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#449475
commutator, Frattini, cosocle	C₂⁴×Q₈	C₂⁴
prime radical	C₂⁴×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁵×Q₈ ◁ G
LECS	G ▷ C₂⁴×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

115738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^11|a^2=k,c^2=i,d^2=i,e^2=k,f^2=k,g^2=k,h^2=k,c^a=ce,c^b=cehk,d^a=defk,d^b=defg,d^c=di,e^a=ek,e^d=ej,f^c=fj,f^d=fj,f^e=fk,g^a=gk,g^c=gjk,g^e=gk,g^f=gk,h^a=hk,h^d=hjk,h^f=hk,h^g=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72267891
maximal subgroup of	#72267891
maximal normal subgroup of	#72267891
maximal quotient of	#72267906

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