#72249481

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#5020751
commutator, Frattini, cosocle	C₂⁶	C₂⁵
prime radical	#10586796	C₂²
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

At least 6438 conjugacy classes of subgroups; only the 582 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|a^2=h,b^2=hi,c^2=ijk,d^2=fgi,e^2=jk,c^a=chik,c^b=ch,d^a=df,d^b=dg,d^c=dgij,e^a=egij,e^b=efghij,e^c=efghi,e^d=efhj,f^a=fj,f^b=fk,f^c=fj,f^d=fjk,f^e=fk,g^a=gk,g^b=gjk,g^c=gk,g^d=gj,g^e=gjk,h^d=hj,h^e=hk,i^d=ik,i^e=ijk

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249531
maximal normal subgroup of	#72249531

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