#72262720

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72241287
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

18890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 32

Polycyclic presentation:

2^11|b^2=jk,c^2=eg,h^2=k,b^a=bjk,c^a=ck,c^b=ce,d^a=def,d^b=dfgh,d^c=dfgij,e^c=ej,e^d=ei,f^b=fjk,f^c=fj,f^d=fi,g^b=gj,g^c=gj,g^d=gik,h^a=hjk,h^b=hk,h^c=hik,h^e=hk,i^b=ik,i^c=ik,j^d=jk

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262722, #72262810
maximal normal subgroup of	#72262722, #72262810

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