#72258552

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#550 ⋊ C₄
Sylow	C₅² ⋊ #34313

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×#1043
commutator, socle	C₂²×C₅²	C₂⁴×C₄
Fitting	C₅²×#550	C₄
prime radical	C₅⋊D₅×#550	C₂
cosocle	C₂²×C₅⋊D₅	C₂⁵
hypercenter	#550	#1043
hypocenter	C₅²	#34313
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #550

13104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 800

Polycyclic presentation:

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

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