#72221552

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#3151 ⋊ C₄
Sylow	C₅ ⋊ #6382816

Subgroups

	subgroup	G/H
center	C₂	C₂×#10617369
commutator	C₂×C₄×C₅	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×AGL(1,𝔽₅)
Fitting	C₅×#3151	C₄
socle	C₂×C₅	C₂×#2366
prime radical	D₅×#3151	C₂
cosocle	C₂×C₄×D₅	C₂⁵
hypercenter	#3151	AGL(1,𝔽₅)
hypocenter	C₅	#6382816
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3151

5708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,5|a^2=f,d^2=g,e^2=h,h^2=i,b^a=bh,c^b=cg,d^a=dh,d^b=dg,d^c=dg,e^a=ehi,e^b=eh,e^c=ehi,g^a=gi,h^a=hi,h^b=hi,h^c=hi,j^a=j^3,j^f=j^4

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#61110
maximal normal subgroup of	AGL(1,𝔽₅)×#61110

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