#72242948

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×#120 ⋊ C₄
Sylow	C₅ ⋊ #7626202

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂⁵×C₄
Fitting	C₂²×C₅×#120	C₄
prime radical	C₂²×D₅×#120	C₂
cosocle	C₂²×D₅	C₂⁶
hypercenter	C₂²×#120	AGL(1,𝔽₅)
hypocenter	C₅	#7626202
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×#120

34564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|c^2=h,d^2=g,d^a=di,d^b=dh,e^c=eh,f^c=fi,j^d=j^3,j^g=j^4

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#62817, #72243209, #72243373, …
maximal normal subgroup of	AGL(1,𝔽₅)×#62817, #72243209, #72243373, …

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