#72225909

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#128069
commutator	C₂²×C₂²⋊C₄	C₂×C₄²
Frattini, cosocle	C₂×#3490	C₂³
prime radical	C₂×#62817	C₂²
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂⁵ ◁ C₂×#62817 ◁ G
LECS	G ▷ C₂×#3490 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

15000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2)(3,4)(6,10,12,17)(11,16)(13,19)(18,20);(5,6,8,11,9,12,15,18)(7,10,13,16,14,17,19,20);(2,4)(5,7)(6,10)(8,13)(9,14)(11,16)(12,17)(15,19)(18,20)

Polycyclic presentation:

2^11|a^2=f,b^2=d,d^2=k,e^2=j,b^a=be,c^a=cg,d^a=dh,e^a=ei,e^b=ehjk,e^c=ej,f^b=fij,f^d=fj,g^b=gj,g^d=gk,h^a=hj,h^b=hk,h^c=hk,h^e=hk,i^b=ijk,i^d=ik,j^b=jk

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226387, #72310881, #72317263
maximal normal subgroup of	#72226387, #72310881, #72317263
maximal quotient of	#72225910, #72226365, #72310507

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