#72243063

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#3484 ⋊ C₄
Sylow	C₅ ⋊ C₂×#12707

Subgroups

	subgroup	G/H
center	C₂	D₅⋊#1781
commutator	C₂²×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#3484	C₄
socle	C₂×C₅	C₂×#1781
prime radical	D₅×#3484	C₂
cosocle	C₂²×D₂₀	C₂⁴
hypercenter	#3484	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#12707
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #3484

8536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|a^2=h,b^2=e,g^2=i,b^a=bg,c^a=ci,c^b=cf,d^a=dhi,e^a=eh,f^a=fi,g^b=ghi,g^c=gi,g^d=gi,g^e=gi,h^b=hi,j^b=j^3,j^e=j^4

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#12707, #72243091, #72243350
maximal normal subgroup of	AGL(1,𝔽₅)×#12707, #72243091, #72243350

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