#72225545

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#495812
commutator	C₄×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	#72225449	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^11|c^2=f,d^2=gi,e^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=be,c^a=ck,c^b=cik,d^a=dh,d^b=dg,d^c=dk,e^a=eh,e^b=eh,e^c=ek,f^b=fk,g^a=gj,g^b=gi,h^a=hj,h^b=hj,i^a=ik,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227164
maximal normal subgroup of	#72227164
maximal quotient of	#72225549

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