#72265091

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#14715
commutator, Frattini, cosocle	C₂⁵×C₄	C₂⁵
LCS	G ▷ C₂⁵×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂⁵×C₄ ▷ C₂³ ▷ C₁

At least 31133 conjugacy classes of subgroups; only the 1967 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|b^2=k,c^2=k,e^2=g,g^2=l,d^a=di,e^a=eh,e^b=eg,e^c=el,e^d=ef,f^a=fj,g^b=gl,h^d=hk,i^e=ijk

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.11);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);