#72216582

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#8231
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×C₈²	C₂³
prime radical	C₂×#1712	C₂×C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #94857 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

1210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|a^2=f,b^2=d,c^2=dg,d^2=g,e^2=hik,f^2=ij,h^2=j,i^2=k,b^a=be,c^a=chik,d^a=djk,e^a=eh,e^b=ehik,e^c=ejk,f^b=fij,f^c=fk,h^a=hj,h^b=hj,i^b=ik

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225289, #72227290
maximal normal subgroup of	#72225289, #72227290
maximal quotient of	#72216585, #72227242

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