#72242118

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ C₂×#12068

Subgroups

	subgroup	G/H
center	C₄	C₂×#65731
commutator	C₅×C₈	C₂³×C₈
Frattini	C₂×C₈	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#2179	C₄
socle	C₂×C₅	C₂×#1580
prime radical	D₅×#508	C₄
cosocle	C₈×Dic₅	C₂⁴
hypercenter	#2179	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#12068
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^9,5|a^2=e,b^2=efgh,c^2=fh,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bf,c^a=cfh,c^b=chi,e^b=ei,e^c=ei,f^a=fh,f^b=fi,h^a=hi,j^a=j^3,j^b=j^3,j^d=j^4,j^e=j^4

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#12068, #72242275, #72242290
maximal normal subgroup of	AGL(1,𝔽₅)×#12068, #72242275, #72242290

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