#72303283

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂³×B(4,𝔽₂)
commutator	C₂×D₄	C₂⁵×C₄
Frattini, cosocle	C₂×#41	C₂⁶
socle	C₂	C₂³×#1889
prime radical	C₂×#10574874	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂²×#585 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

Only the 10998 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|b^2=ik,d^2=g,g^2=k,h^2=k,b^a=bi,d^a=dk,d^b=dh,d^c=dk,e^c=ek,e^d=ek,f^d=fk,g^b=gk,h^a=hj,h^b=hj,i^d=ijk,i^h=ik,j^b=jk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303611
maximal normal subgroup of	#72303611

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