#72262736

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#2403
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁵
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

At least 8188 conjugacy classes of subgroups; only the 2528 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|c^2=gi,e^2=fgh,f^2=k,c^a=cg,c^b=cj,e^a=egh,e^b=egh,e^c=ehi,e^d=ef,f^d=fk,g^e=gk,h^c=hk,h^e=hk,j^e=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262791
maximal normal subgroup of	#72262791

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