#72237873

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235808
commutator, Frattini, cosocle	#2308	C₂⁴
LCS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#2308 ◁ G
LECS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

6360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 32

Polycyclic presentation:

2^11|a^2=ijk,c^2=ehi,e^2=ik,f^2=ijk,b^a=bf,c^a=ce,c^b=cfg,d^b=dijk,d^c=dik,e^a=ei,e^b=eh,e^c=ejk,f^a=fijk,f^b=fijk,f^c=fh,f^e=fj,g^a=gijk,g^c=gijk,g^e=gk,h^a=hj,h^c=hjk,h^e=hk,h^f=hk,i^b=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237931, #72237974, #72237998, …
maximal normal subgroup of	#72237931, #72237974, #72237998, …

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