#72258376

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1789 ⋊ C₄
Sylow	C₅ ⋊ #392580

Subgroups

	subgroup	G/H
center	C₂²	C₂×#65699
commutator	C₂²×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#1789	C₄
socle	C₂²×C₅	C₂×#385
prime radical	C₂²×D₅×#41	C₂²
cosocle	C₂²×C₄×D₅	C₂⁴
hypercenter	#1789	AGL(1,𝔽₅)
hypocenter	C₅	#392580
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1789

7724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^9,5|b^2=ghi,c^2=e,g^2=i,h^2=i,b^a=bg,c^b=cghi,d^b=df,g^a=gi,g^c=gi,h^c=hi,j^c=j^3,j^e=j^4

Conjugacy classes & character table

136 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, 2, 5]);;
gap> SetPower(coll, 2, F.7*F.8*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);