#72267676

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂⁴×Q₈	C₂⁴
prime radical	C₂⁴×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ C₂⁵×Q₈ ◁ G
LECS	G ▷ C₂⁴×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

116205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 512

Polycyclic presentation:

2^11|b^2=g,c^2=gh,e^2=k,f^2=k,b^a=be,c^a=cf,c^b=cgi,d^a=dk,d^b=dghk,d^c=dh,e^a=ek,e^b=ek,e^c=ej,f^a=fk,f^b=fjk,f^c=fk,f^e=fk,j^a=jk

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267688
maximal normal subgroup of	#72267688

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