#72258455

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₅² ⋊ C₄²
Sylow	C₅² ⋊ C₄×#379

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)²
commutator, socle	C₂²×C₅²	C₂²×C₄²
Fitting	C₂⁴×C₅²	C₄²
prime radical	C₂⁴×D₅²	C₂²
cosocle	C₂²×D₅²	C₂⁴
hypercenter	C₂⁴	AGL(1,𝔽₅)²
hypocenter	C₅²	C₄×#379
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂⁴

14149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^8,5^2|a^2=e,b^2=f,c^a=cg,d^a=dgh,i^a=i^2,i^b=i^3,i^e=i^4,i^f=i^4,j^a=i^4j,j^b=j^3,j^e=i^2j,j^f=j^4

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258468
maximal normal subgroup of	#72258468

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