#72242730

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×#120 ⋊ C₄
Sylow	C₅ ⋊ C₂×#12687

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₅⋊#99
commutator	C₂⁴×C₅	C₂³×C₄
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₅×#120	C₄
socle	C₂²×C₅	C₂²×#99
prime radical	C₂²×D₅×#120	C₂
cosocle	C₂²×C₂²⋊D₅	C₂⁴
hypercenter	C₂²×#120	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#12687
LCS	G ▷ C₂⁴×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂²×#120

22592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|a^2=e,e^2=i,b^a=bf,c^a=cg,e^b=eh,e^c=ehi,f^a=fh,f^e=fi,g^a=ghi,g^e=gi,h^a=hi,j^a=j^3,j^d=j^4,j^e=j^4

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#12687, #72242884, #72242934, …
maximal normal subgroup of	AGL(1,𝔽₅)×#12687, #72242884, #72242934, …

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, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, 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);