#72243287

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×#120 ⋊ C₄
Sylow	C₅ ⋊ C₂×#8541

Subgroups

	subgroup	G/H
center	C₂²	C₂×#65800
commutator	C₂⁴×C₅	C₂³×C₄
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₅×#120	C₄
socle	C₂²×C₅	C₂×#379
prime radical	C₂²×D₅×#120	C₂
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	C₂²×#120	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#8541
LCS	G ▷ C₂⁴×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂²×#120

23164 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 23164 conjugacy classes of subgroups — too many to draw. Only its 462 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^9,5|c^2=e,c^a=cg,c^b=cf,d^a=di,d^b=dh,e^a=ei,e^b=eh,f^c=fh,g^c=gi,j^c=j^3,j^e=j^4

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#8541, #72243377, #72243444
maximal normal subgroup of	AGL(1,𝔽₅)×#8541, #72243377, #72243444

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