#72221590

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ C₂×#11992

Subgroups

	subgroup	G/H
center	C₂	D₅⋊#1555
commutator	C₄²×C₅	C₂³×C₄
Frattini	C₄²	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#3151	C₄
socle	C₂×C₅	C₂×#1555
prime radical	D₅×#3151	C₂
cosocle	C₄×#168	C₂⁴
hypercenter	#3151	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#11992
LCS	G ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3151

2348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,5|a^2=e,d^2=h,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bf,c^a=cfh,c^b=cghi,d^b=dh,d^c=dhi,e^b=eh,e^c=ehi,f^a=fg,f^b=fgh,f^c=fgh,g^a=gi,g^b=gi,g^c=gi,h^b=hi,h^c=hi,j^a=j^3,j^e=j^4

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11992
maximal normal subgroup of	AGL(1,𝔽₅)×#11992

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