#72223375

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#535 ⋊ C₂²
Sylow	C₅² ⋊ C₂×#2008

Subgroups

	subgroup	G/H
center	C₂	#72223346
commutator, cosocle	C₂²×C₄×C₅²	C₂⁴
Frattini	C₂²×C₄	C₂²×D₅²
Fitting	C₅²×#535	C₂²
socle	C₂×C₅²	C₂×#392
hypercenter	#535	D₅²
hypocenter	C₅²	C₂×#2008
LCS	G ▷ C₂²×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

6132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,5^2|a^2=f,f^2=h,b^a=bfh,c^a=cfg,c^b=ce,e^a=eh,f^b=fh,f^c=fh,g^b=gh,i^a=i^4,i^d=i^4,j^a=i^3j,j^d=j^4

Conjugacy classes & character table

226 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.6);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);