#72242680

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#2875

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅²
commutator	C₂×C₅²	C₂³×C₄²
Fitting	C₄×C₅²×#41	C₂²
socle	C₂²×C₅²	C₂⁴×C₄
prime radical	C₂×#41×C₅⋊D₅	C₂²
cosocle	C₂×C₄×C₅²	C₂⁵
hypercenter	C₄×#41	D₅²
hypocenter	C₅²	C₂×#2875
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

8284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 320

Polycyclic presentation:

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

Conjugacy classes & character table

568 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.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 9, 3, F.9^4);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> SetConjugate(coll, 10, 3, F.9^4*F.10);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);