#72223047

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×#41 ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×#585

Subgroups

	subgroup	G/H
center	C₄	C₂³×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₂⁵×C₄
Frattini	C₂	C₂⁴×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×#41	C₂×C₄
prime radical	#72220977	C₂
cosocle	C₂×C₅⋊D₅	C₂⁶
hypercenter	C₂×#41	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#585
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×#41

24420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

244 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, 2, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 5, F.9^3);;
gap> SetConjugate(coll, 9, 7, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 5, F.10^3);;
gap> SetConjugate(coll, 10, 7, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);