#72227238

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#1776
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×C₈²	C₂³
socle, prime radical	C₂³	#8276
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #94857 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 2048

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,6,5,7,8,3)(9,10,12,16,14,18,21,19)(11,15,22,24,13,23,20,17)(25,26,28,27,29,33,37,35,31,34,30,36,38,32,39,40);(10,15)(11,20)(12,21)(13,22)(16,17)(18,23)(19,24)(25,27,30,33,29,35,39,34,31,36,28,32,38,40,37,26);(1,3,8,7,5,6,4,2)(9,11,21,20,14,13,12,22)(10,15,19,17,18,23,16,24)(26,32,34,33)(27,35,36,40)(28,30)(37,39)

Polycyclic presentation:

2^11|a^2=d,b^2=f,c^2=dfg,d^2=g,e^2=hk,f^2=i,g^2=k,h^2=j,i^2=k,b^a=be,c^a=ci,c^b=chijk,d^b=djk,d^c=dk,e^a=eh,e^b=ehj,e^c=ej,f^a=fk,f^c=fk,h^a=hj,h^b=hj

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227290, #72258190
maximal normal subgroup of	#72227290, #72258190
maximal quotient of	#72227239

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