#72227249

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#9221
commutator	C₈²	C₂²×C₈
Frattini, cosocle	C₄×C₈²	C₂³
prime radical	C₂×#12424	C₄
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

1278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^11|a^2=d,d^2=g,e^2=hi,f^2=i,g^2=k,h^2=j,i^2=k,b^a=bf,c^a=ce,c^b=cfk,d^b=di,d^c=dijk,e^a=eh,e^b=ei,e^c=ehijk,f^a=fk,f^b=fik,f^c=fik,g^b=gk,g^c=gk,h^a=hj,h^c=hj,i^b=ik,i^c=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227290
maximal normal subgroup of	#72227290
maximal quotient of	#72227250, #72227270

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