#72253157

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #12067

Subgroups

	subgroup	G/H
center	C₂²	#13104743
commutator	C₅²×C₈	C₂²×C₈
Frattini	C₂×C₈	C₂²×#1043
Fitting	C₅²×#357	C₄
socle	C₂²×C₅²	#442
prime radical	C₂×#11716693	C₄
cosocle	C₈×#1039	C₂³
hypercenter	#357	#1043
hypocenter	C₅²	#12067
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

984 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 3200

Polycyclic presentation:

2^8,5^2|a^2=d,b^2=df,c^2=e,d^2=f,e^2=g,g^2=h,c^a=ce,c^b=cg,d^c=dh,e^a=eg,e^b=eh,g^a=gh,i^a=i^3,i^b=i^3,i^c=i^4,i^d=i^4,j^a=j^3,j^b=j^3,j^c=j^4,j^d=j^4

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253165
maximal normal subgroup of	#72253165

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