#72243222

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1781 ⋊ C₄
Sylow	C₅ ⋊ #106132

Subgroups

	subgroup	G/H
center	C₂²	#10604252
commutator	C₂⁴×C₅	C₂×C₄²
Frattini	C₂⁵	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#1781	C₄
socle	C₂²×C₅	C₂²⋊#95
prime radical	C₂³×D₄×D₅	C₂²
cosocle	C₂⁵×D₅	C₂³
hypercenter	#1781	AGL(1,𝔽₅)
hypocenter	C₅	#106132
LCS	G ▷ C₂⁴×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #1781

7028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|b^2=e,c^2=d,b^a=bg,c^a=cg,c^b=cf,d^a=di,d^b=dh,e^a=ei,e^c=eh,f^b=fh,f^c=fh,g^b=gi,g^c=gi,j^c=j^3,j^d=j^4

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243234, AGL(1,𝔽₅)×#8541, #72243462
maximal normal subgroup of	#72243234, AGL(1,𝔽₅)×#8541, #72243462

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