#72259481

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂³×D₄
commutator	C₂³×C₄	C₂⁴×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁵
socle	C₂⁴	C₂³×C₂²⋊C₄
prime radical	#72258267	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×#379 ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

135727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 256

Polycyclic presentation:

2^11|b^2=fgk,e^2=k,f^2=k,g^2=k,b^a=bf,c^a=ci,c^b=ch,d^a=dhk,d^b=dhi,e^a=efgjk,e^b=efk,e^d=ek,f^a=fk,f^e=fk,j^a=jk,j^e=jk

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259591
maximal normal subgroup of	#72259591

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