#72223517

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×H(5,𝔽₂) ⋊ C₄
Sylow	C₅² ⋊ #60087

Subgroups

	subgroup	G/H
center	C₂	C₂³×C₂²⋊#1044
commutator	C₄×C₅²	C₂⁴×C₄
Frattini	C₄	C₂⁴×#1044
Fitting	C₂×C₅²×H(5,𝔽₂)	C₄
socle	C₂×C₅²	C₂³×C₂²⋊C₄
prime radical	C₂×H(5,𝔽₂)×C₅⋊D₅	C₂
cosocle	C₄×C₅⋊D₅	C₂⁵
hypercenter	C₂×H(5,𝔽₂)	#1044
hypocenter	C₅²	#60087
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

16868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

286 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, 2, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 6, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10^3);;
gap> SetConjugate(coll, 10, 6, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);