#72307309

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#114109
commutator	C₂²×C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂²×#405	C₂³
prime radical	#72237269	C₂
LCS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^11|b^2=f,c^2=dej,d^2=i,e^2=jk,g^2=j,j^2=k,b^a=bf,c^a=cfgjk,c^b=cd,d^b=dk,d^c=dh,e^a=ej,e^b=ehik,e^c=eh,e^d=ek,f^c=fik,g^a=gjk,g^b=gik,g^c=gik,h^c=hk,j^a=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307393, #72308088
maximal normal subgroup of	#72307393, #72308088

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