#72221572

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#2165 ⋊ C₄
Sylow	C₅ ⋊ #126894

Subgroups

	subgroup	G/H
center	C₄	#10604379
commutator	C₂×C₅×C₈	C₂×C₄²
Frattini	C₄×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#2165	C₄
socle	C₂×C₅	#7904
prime radical	C₂³×D₅	#104
cosocle	C₄×C₈×D₅	C₂³
hypercenter	#2165	AGL(1,𝔽₅)
hypocenter	C₅	#126894
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×M₁₆ ◁ #2165

1026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,5|a^2=d,b^2=deg,c^2=g,e^2=g,f^2=hi,h^2=i,b^a=bf,c^a=ch,c^b=cg,e^a=eh,f^a=fh,f^c=fi,g^a=gi,h^a=hi,j^a=j^3,j^b=j^3,j^d=j^4

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11669
maximal normal subgroup of	AGL(1,𝔽₅)×#11669

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