#72269856

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×B(4,𝔽₂) ⋊ C₄
Sylow	C₅ ⋊ #364532

Subgroups

	subgroup	G/H
center	C₂	#13028194
commutator	C₂⁴×C₅	C₂³×C₄
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₅×B(4,𝔽₂)	C₄
socle	C₂×C₅	#13967
prime radical	C₂×D₅×B(4,𝔽₂)	C₂
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	C₂×B(4,𝔽₂)	AGL(1,𝔽₅)
hypocenter	C₅	#364532
LCS	G ▷ C₂⁴×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×B(4,𝔽₂)

6368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|a^2=i,b^2=e,b^a=bf,c^b=ch,d^a=dg,d^c=dh,e^a=ei,f^b=fi,f^c=fi,g^c=gi,h^a=hi,j^b=j^3,j^c=j^4,j^e=j^4

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#23996, #72269870
maximal normal subgroup of	AGL(1,𝔽₅)×#23996, #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.9);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, 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, 2, F.10^3);;
gap> SetConjugate(coll, 10, 3, F.10^4);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);