#72242710

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×AΣL(1,𝔽₁₆) ⋊ C₄
Sylow	C₅ ⋊ #146661

Subgroups

	subgroup	G/H
center	C₂²	AGL(1,𝔽₅)×#99
commutator	C₂⁴×C₅	C₂×C₄²
Frattini	C₂²×D₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×AΣL(1,𝔽₁₆)	C₄
socle	C₂²×C₅	C₄×#99
prime radical	C₂×D₅×#120	C₂²
cosocle	C₂²×D₄×D₅	C₂³
hypercenter	C₂×AΣL(1,𝔽₁₆)	AGL(1,𝔽₅)
hypocenter	C₅	#146661
LCS	G ▷ C₂⁴×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×AΣL(1,𝔽₁₆)

5000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 17 points:

(2,3,4,5)(10,12)(13,17);(1,2,5,3,4)(7,9)(11,14)(15,16);(6,7)(8,9)(10,11)(12,15,17,14)(13,16)

Polycyclic presentation:

2^9,5|a^2=d,c^2=e,b^a=bf,c^a=cg,d^b=dh,d^c=dhi,f^a=fh,f^d=fi,g^a=ghi,g^d=gi,h^a=hi,j^c=j^3,j^e=j^4

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#12687, #72242889, #72242920
maximal normal subgroup of	AGL(1,𝔽₅)×#12687, #72242889, #72242920

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.4);;
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, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*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);