#72223239

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₅×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#1721 ⋊ C₂
Sylow	C₅² ⋊ #19745

Subgroups

	subgroup	G/H
center	C₂²	#127×C₅⋊D₅
commutator, cosocle	C₄²×C₅²	C₂⁴
Frattini	C₄²	C₂³×C₅⋊D₅
Fitting	C₅²×#1721	C₂
socle	C₂²×C₅²	C₂×#127
hypercenter	#1721	C₅⋊D₅
hypocenter	C₅²	#19745
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1721

3976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,5^2|a^2=e,c^2=fh,d^2=f,e^2=g,f^2=h,b^a=be,c^a=cg,c^b=cf,d^b=dfh,d^c=dh,e^b=eg,f^b=fh,i^d=i^4,j^d=j^4

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#19745
maximal normal subgroup of	C₅⋊D₅×#19745

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