#72281807

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#1889
commutator	C₂×#41	C₂²×C₄²
Frattini, cosocle	C₂³×#41	C₂⁴
socle	C₂	#72280434
prime radical	C₂²×#3025	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×#3428 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

At least 10732 conjugacy classes of subgroups; only the 1398 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|b^2=ik,c^2=g,d^2=f,e^2=k,f^2=k,h^2=k,b^a=bi,c^a=ck,d^b=dh,d^c=de,f^b=fk,f^c=fk,g^d=gk,h^a=hj,h^b=hj,i^d=ijk,i^h=ik,j^b=jk

Conjugacy classes & character table

188 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, 2, F.9*F.11);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);