#72221643

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×D₄ ⋊ C₄²
Sylow	C₅² ⋊ C₄×#418

Subgroups

	subgroup	G/H
center	C₂	C₂×#13107468
commutator	C₄×C₅²	C₂²×C₄²
Frattini	C₄	C₂²×AGL(1,𝔽₅)²
Fitting	C₂×C₅²×D₄	C₄²
socle	C₂×C₅²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×D₄×D₅²	C₂²
cosocle	C₄×D₅²	C₂⁴
hypercenter	C₂×D₄	AGL(1,𝔽₅)²
hypocenter	C₅²	C₄×#418
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

4249 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221730
maximal normal subgroup of	#72221730

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