#72242155

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#2263 ⋊ C₄
Sylow	C₅ ⋊ C₂×#13658

Subgroups

	subgroup	G/H
center	C₂	C₂×#10604121
commutator	C₅×C₁₆	C₂³×C₄
Frattini	C₁₆	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#2263	C₄
socle	C₂×C₅	C₂×#2177
prime radical	D₅×#2263	C₂
cosocle	D₅×C₁₆	C₂⁴
hypercenter	#2263	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#13658
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2263

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^9,5|a^2=f,c^2=e,f^2=gh,g^2=hi,h^2=i,b^a=bfi,c^a=chi,d^a=di,e^a=ei,f^b=fg,f^c=fi,g^b=gh,h^b=hi,j^c=j^3,j^e=j^4

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#13658
maximal normal subgroup of	AGL(1,𝔽₅)×#13658

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, 3, F.5);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, 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);