#72223142

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₅²×M₁₆ ⋊ C₄
Sylow	C₅² ⋊ #20445

Subgroups

	subgroup	G/H
center	C₄	C₂²×C₄⋊#1044
commutator	C₄×C₅²	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#1044
Fitting	C₄×C₅²×M₁₆	C₄
socle	C₂×C₅²	C₂×C₄×C₄⋊C₄
prime radical	C₂³×C₅⋊D₅	#41
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₄×M₁₆	#1044
hypocenter	C₅²	#20445
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₄×M₁₆

2740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

328 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226402
maximal normal subgroup of	#72226402

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