#72258296

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393929
commutator, Frattini, cosocle	C₂⁵×D₄	C₂⁴
LCS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ C₂⁴×#120 ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

At least 12914 conjugacy classes of subgroups; only the 531 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|c^2=l,b^a=bh,c^a=cg,c^b=cf,d^b=di,d^c=de,e^b=ei,f^a=fj,f^d=fi,g^b=gjk,g^f=gl,h^c=hkl,j^c=jl

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258531, #72265847
maximal normal subgroup of	#72265847
maximal quotient of	#72266691

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