#72322158

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂³×D₄
commutator	C₂³×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁵
socle	C₂⁴	C₂²×C₄×D₄
prime radical	C₂⁵×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×#407 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

46244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 256

Polycyclic presentation:

2^11|a^2=f,d^2=fj,f^2=j,i^2=k,b^a=bi,d^a=dh,d^b=dg,d^c=dgk,e^d=ej,i^a=ik,i^b=ik

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322335
maximal normal subgroup of	#72322335

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