#72221559

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#3184 ⋊ C₄
Sylow	C₅ ⋊ #495541

Subgroups

	subgroup	G/H
center	C₂	#13035991
commutator	C₂×C₅×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#3184	C₄
socle	C₂×C₅	#21395
prime radical	D₅×#3184	C₂
cosocle	C₂×C₈×D₅	C₂⁴
hypercenter	#3184	AGL(1,𝔽₅)
hypocenter	C₅	#495541
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3184

1930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#33688
maximal normal subgroup of	AGL(1,𝔽₅)×#33688

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