#72269225

order	12800
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#7333

Subgroups

	subgroup	G/H
center	C₄	#72242198
commutator	C₅²×C₈	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×#346	C₂×C₄
socle	C₂×C₅²	C₂×#1313
prime radical	C₂²×C₅⋊D₅	#347
cosocle	#13099536	C₂³
hypercenter	#346	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₂×#7333
LCS	G ▷ C₅²×C₈ ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ #346

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^9,5^2|a^2=e,b^2=d,c^2=g,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bghi,c^a=ci,c^b=ch,d^a=dh,d^c=di,e^b=eh,e^d=ei,f^a=fi,g^b=gi,j^a=j^2,j^b=j^3,j^d=j^4,j^e=j^4,k^a=j^3k^3,k^b=k^3,k^d=k^4,k^e=k^4

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269242
maximal normal subgroup of	#72269242

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> SetConjugate(coll, 11, 1, F.10^3*F.11^3);;
gap> SetConjugate(coll, 11, 2, F.11^3);;
gap> SetConjugate(coll, 11, 4, F.11^4);;
gap> SetConjugate(coll, 11, 5, F.11^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);