#72258570

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×#120 ⋊ C₄
Sylow	C₅² ⋊ #14694

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×#1043
commutator, socle	C₂³×C₅²	C₂³×C₄
Fitting	C₂×C₅²×#120	C₄
prime radical	C₂×#120×C₅⋊D₅	C₂
cosocle	C₂³×C₅⋊D₅	C₂⁴
hypercenter	C₂×#120	#1043
hypocenter	C₅²	#14694
LCS	G ▷ C₂³×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂×#120

28136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 400

Polycyclic presentation:

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

Conjugacy classes & character table

256 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, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);