#72253762

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#3484 ⋊ C₄
Sylow	C₅ ⋊ C₂×#21776

Subgroups

	subgroup	G/H
center	C₂	C₂×#10617506
commutator	C₂³×C₅	C₂⁴×C₄
Frattini	C₂³	C₂⁴×AGL(1,𝔽₅)
Fitting	C₅×#3484	C₄
socle	C₂×C₅	C₂×#2308
prime radical	D₅×#3484	C₂
cosocle	C₂³×D₅	C₂⁵
hypercenter	#3484	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#21776
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #3484

10544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|a^2=f,b^a=bh,d^a=dg,d^c=dhi,e^b=ei,f^b=fi,g^c=gi,h^a=hi,j^a=j^3,j^f=j^4

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253792, AGL(1,𝔽₅)×#21776, #72269870
maximal normal subgroup of	#72253792, AGL(1,𝔽₅)×#21776, #72269870

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