#72253164

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#509 ⋊ C₄
Sylow	C₅² ⋊ C₂×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center	C₂	C₂×#13104743
commutator	C₅²×C₈	C₂³×C₄
Frattini	C₈	C₂³×#1043
Fitting	C₅²×#509	C₄
socle	C₂×C₅²	C₂×#442
prime radical	C₅⋊D₅×#509	C₂
cosocle	C₈×C₅⋊D₅	C₂⁴
hypercenter	#509	#1043
hypocenter	C₅²	C₂×AGL(1,ℤ/16)
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

2840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 400

Polycyclic presentation:

2^8,5^2|a^2=e,b^2=e,c^2=f,f^2=g,g^2=h,c^a=cf,c^b=cg,e^c=eh,f^a=fg,f^b=fh,g^a=gh,i^a=i^3,i^b=i^3,i^d=i^4,i^e=i^4,j^a=j^3,j^b=j^3,j^d=j^4,j^e=j^4

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×AGL(1,ℤ/16)
maximal normal subgroup of	#1043×AGL(1,ℤ/16)
maximal quotient of	#72253165

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