#72253602

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#23900 ⋊ C₂
Sylow	C₅ ⋊ #6499874

Subgroups

	subgroup	G/H
center	C₂	#12278359
commutator, cosocle	C₂⁴×C₅	C₂⁵
Frattini	C₂⁴	C₂⁴×D₅
Fitting	C₅×#23900	C₂
socle	C₂×C₅	#35346
hypercenter	#23900	D₅
hypocenter	C₅	#6499874
LCS	G ▷ C₂⁴×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #23900

9160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|a^2=i,d^a=dg,d^b=dh,e^a=ef,e^c=eh,f^c=fi,g^b=gi,h^a=hi,j^b=j^4

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#6499874
maximal normal subgroup of	D₅×#6499874

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