#72236986

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#6628896
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #35527 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

42866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^11|a^2=ik,e^2=j,f^2=k,c^a=cf,c^b=cg,d^a=dik,e^c=eh,e^d=eik,f^b=fi,f^c=fk,f^d=fk,g^a=gik,g^e=gj,g^f=gk,h^b=hj,h^d=hk,i^c=ik

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237014, #72237019, #72241341, …
maximal normal subgroup of	#72237014, #72237019, #72241341, …
maximal quotient of	C₂×#72236986
automorphism group of	#1911, #1912, #3019, …

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