#72253173

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×C₈×D₄ ⋊ C₄
Sylow	C₅² ⋊ #8200

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#1043
commutator	C₄×C₅²	C₂×C₄×C₈
Frattini	C₄²	C₂²×#1043
Fitting	C₅²×C₈×D₄	C₄
socle	C₂²×C₅²	C₄×C₂²⋊C₄
prime radical	C₂×D₄×C₅⋊D₅	C₂×C₄
cosocle	C₄²×C₅⋊D₅	C₂³
hypercenter	C₈×D₄	#1043
hypocenter	C₅²	#8200
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₄

2360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1600

Polycyclic presentation:

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

Conjugacy classes & character table

328 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×#1580
maximal normal subgroup of	#1043×#1580
maximal quotient of	#72253183

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