#72243114

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#23895 ⋊ C₂
Sylow	C₅ ⋊ #511790

Subgroups

	subgroup	G/H
center	C₂	#12248070
commutator, cosocle	C₂²×C₅×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₅
Fitting	C₅×#23895	C₂
socle	C₂×C₅	#32898
hypercenter	#23895	D₅
hypocenter	C₅	#511790
LCS	G ▷ C₂²×C₅×D₄ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #23895

6038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|a^2=i,d^2=e,g^2=i,b^a=bh,c^a=ci,d^a=df,d^b=dfgi,d^c=de,e^a=ei,f^b=fh,f^c=fi,f^d=fi,g^a=gh,g^b=gh,g^c=gi,g^d=gi,g^f=gi,h^d=hi,j^b=j^4

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#511790
maximal normal subgroup of	D₅×#511790

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