#72338598

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1781
commutator	C₂×C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×#117	C₂⁵
prime radical	C₂×#7737411	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×#3498 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

67724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^11|a^2=jk,b^2=jk,c^2=fg,f^2=j,h^2=k,c^a=cf,d^c=djk,e^a=ejk,e^b=ejk,e^c=eh,f^a=fj,f^c=fi,f^e=fj,g^a=gi,g^c=gi,g^e=gk,g^f=gk,h^a=hj,h^c=hj,h^e=hk,i^c=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338614, #72338615, #72340198
maximal normal subgroup of	#72338614, #72338615, #72340198

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