#72258273

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10991
commutator	C₂³×D₄	C₂²×C₄
Frattini, cosocle	C₂⁴×D₄	C₂³
prime radical	#6437621	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=df,d^2=j,e^2=j,b^a=bd,c^a=ce,c^b=cefghij,d^a=di,d^b=dh,d^c=di,e^a=eij,e^b=ehi,e^c=ej,f^a=fhij,f^b=fh,f^c=fi,f^d=fj,f^e=fj,g^b=gij,g^c=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258297, #72265936, #72265937
maximal normal subgroup of	#72258297, #72265936, #72265937
maximal quotient of	#72258401, #72266234

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