#72258341

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×#392 ⋊ C₄
Sylow	C₅ ⋊ #155116

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂⁴×C₅	C₂³×C₄
Fitting	C₂×C₅×#392	C₄
prime radical	C₂×D₅×#392	C₂
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	C₂×#392	AGL(1,𝔽₅)
hypocenter	C₅	#155116
LCS	G ▷ C₂⁴×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂×#392

14620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 320

Polycyclic presentation:

2^9,5|a^2=f,c^2=e,d^2=ei,b^a=bg,c^a=cf,d^b=dh,d^c=di,j^c=j^3,j^d=j^3,j^e=j^4

Conjugacy classes & character table

160 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, 2, 5]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);