#72211375

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#1270
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×#1744	C₂³
socle	C₂³	C₂×#1270
prime radical	C₂⁵	#327
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ C₂²×#1744 ◁ G
LECS	G ▷ C₂×#1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁

3824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|a^2=d,c^2=f,e^2=gj,f^2=i,g^2=j,h^2=k,b^a=bj,c^a=ce,e^a=eg,e^c=eh,f^a=fghj,f^e=fk,g^a=gj,g^c=gk,h^a=hk,i^a=ijk

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211417, #72213065, #72213854, …
maximal normal subgroup of	#72213065, #72213854, #72214551, …

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