#72264786

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁶
commutator, socle	C₂⁴	C₂⁵×C₄
prime radical	C₂³×D₄²	C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

125782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 256

Polycyclic presentation:

2^11|a^2=g,d^2=h,e^2=i,h^2=k,b^a=bg,d^a=di,d^b=dj,d^c=dj,e^c=ei,e^d=eg,f^d=fk

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264893
maximal normal subgroup of	#72264893

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