#72223157

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#493 ⋊ C₄
Sylow	C₅² ⋊ #13261

Subgroups

	subgroup	G/H
center	C₂	#72223155
commutator	C₂×C₅²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×#1044
Fitting	C₅²×#493	C₄
socle	C₂×C₅²	#1982
prime radical	C₅⋊D₅×#493	C₂
cosocle	C₂×C₈×C₅⋊D₅	C₂³
hypercenter	#493	#1044
hypocenter	C₅²	#13261
LCS	G ▷ C₂×C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

1892 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 1892 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,5^2|a^2=d,c^2=f,e^2=gh,g^2=h,b^a=be,c^a=cgh,c^b=cf,e^a=eg,e^b=eg,f^a=fh,g^a=gh,g^b=gh,i^a=i^2,i^d=i^4,j^a=i^3j^3,j^d=j^4

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

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