#72266614

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#8541
commutator	C₂²×C₄⋊C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄⋊C₄	C₂⁴
socle	C₂²	C₂×#8541
prime radical	#72266610	C₂
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄⋊C₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

14154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^11|a^2=e,b^2=ik,c^2=egjk,d^2=k,e^2=j,f^2=k,g^2=k,b^a=befk,c^a=cg,c^b=cefhk,d^b=dik,d^c=dhk,e^b=ei,e^c=eh,f^a=fij,f^b=fijk,f^c=fhi,f^d=fk,f^e=fk,g^a=ghk,g^b=gjk,g^c=gh,g^d=gk,g^e=gk,h^a=hk,h^b=hk,h^c=hk,i^a=ik

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266634, #72266649
maximal normal subgroup of	#72266634, #72266649

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