#72319144

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#442
commutator	C₂×C₈	C₂⁵×C₄
Frattini, cosocle	C₄×C₈	C₂⁶
prime radical	#72318881	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #6342482 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

16340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 512

Polycyclic presentation:

2^11|a^2=ik,b^2=g,c^2=ik,d^2=gk,e^2=k,g^2=i,h^2=j,j^2=k,c^b=ch,d^c=djk,e^a=ei,e^b=ei,e^c=ei,f^a=fik,f^b=fk,f^e=fik,g^c=gk,h^b=hj,h^c=hjk,h^d=hk,j^b=jk,j^c=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

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