#72221097

order	6400
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×C₈² ⋊ C₄
Sylow	C₅² ⋊ #7846

Subgroups

	subgroup	G/H
center, Frattini	C₄²	C₂²×#1043
commutator, socle	C₂²×C₅²	C₄³
Fitting	C₅²×C₈²	C₄
prime radical	C₂²×C₅⋊D₅	C₂×C₄²
cosocle	C₄²×C₅⋊D₅	C₂³
hypercenter	C₈²	#1043
hypocenter	C₅²	#7846
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₄² ◁ C₈²

2328 conjugacy classes of subgroups; only the 474 normal subgroups are shown.

normal subgroups only

This group has 2328 conjugacy classes of subgroups — too many to draw. Only its 474 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=e,b^2=d,c^2=f,e^2=g,f^2=h,b^a=bg,c^b=ch,i^b=i^3,i^d=i^4,j^b=j^3,j^d=j^4

Conjugacy classes & character table

496 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×#1455, #72223252, #72223260
maximal normal subgroup of	#1043×#1455, #72223252, #72223260

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