#72242251

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#1398

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂×AGL(1,𝔽₅)²
commutator	C₄×C₅²	C₂×C₄×C₈
Fitting	C₂×C₅²×C₈	C₄²
socle	C₂×C₅²	C₂×#346
prime radical	C₂²×C₅⋊D₅	M₃₂
cosocle	#11716583	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)²
hypocenter	C₅²	C₂×#1398
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₂×C₈

525 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 160

Polycyclic presentation:

2^8,5^2|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,b^a=bg,c^a=ch,d^b=dh,e^a=eh,i^a=i^2,i^b=i^3,i^d=i^4,i^e=i^4,j^a=i^4j,j^b=j^3,j^d=i^2j,j^e=j^4

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242271
maximal normal subgroup of	#72242271

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