#72253129

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#365 ⋊ C₄
Sylow	C₅² ⋊ #12677

Subgroups

	subgroup	G/H
center	C₂	#72220668
commutator	C₅²×C₈	C₂×C₄²
Frattini	C₂×C₈	C₂²×#1043
Fitting	C₅²×#365	C₄
socle	C₂×C₅²	#1776
prime radical	C₂²×C₅⋊D₅	AGL(1,ℤ/8)
cosocle	C₂×C₈×C₅⋊D₅	C₂³
hypercenter	#365	#1043
hypocenter	C₅²	#12677
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 400

Polycyclic presentation:

2^8,5^2|a^2=d,b^2=de,c^2=f,f^2=g,g^2=h,c^a=cf,c^b=cg,d^c=dh,f^a=fg,f^b=fh,g^a=gh,i^b=i^3,i^e=i^4,j^b=j^3,j^e=j^4

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×AGL(1,ℤ/16)
maximal normal subgroup of	#1043×AGL(1,ℤ/16)
maximal quotient of	#72253131

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