#72265660

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#2308
commutator	C₂⁵	C₂⁴×C₄
Frattini, cosocle	C₂⁶	C₂⁵
prime radical	#72217875	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂ ▷ C₁

At least 8863 conjugacy classes of subgroups; only the 1848 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|c^2=g,b^a=bf,d^a=dh,d^c=di,e^b=eik,e^c=ej,g^d=gk,i^c=ik,j^b=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265689, #72265794, #72265836
maximal normal subgroup of	#72265689, #72265794, #72265836
maximal quotient of	#72266093

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