#72242229

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#7333 ⋊ C₂
Sylow	C₅ ⋊ AGL(1,ℤ/32)

Subgroups

	subgroup	G/H
center	C₂	#12011186
commutator	C₅×C₁₆	C₂²×C₈
Frattini	#346	C₂²×D₅
Fitting	C₅×#7333	C₂
socle	C₂×C₅	#12067
prime radical	#10600166	C₄
cosocle	C₅×#346	C₂³
hypercenter	#7333	D₅
hypocenter	C₅	AGL(1,ℤ/32)
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1313 ◁ #7333

276 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^9,5|a^2=e,b^2=d,c^2=df,d^2=f,e^2=g,g^2=h,h^2=i,b^a=beghi,c^a=cghi,d^a=dhi,e^b=eg,e^c=eh,e^d=ei,f^a=fi,g^b=gh,g^c=gi,h^b=hi,j^b=j^4

Conjugacy classes & character table

139 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×AGL(1,ℤ/32)
maximal normal subgroup of	D₅×AGL(1,ℤ/32)

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