#72242825

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1781 ⋊ C₄
Sylow	C₅ ⋊ C₂×#7307

Subgroups

	subgroup	G/H
center	C₂²	C₂×#64866
commutator	C₂²×C₄×C₅	C₂×C₄²
Frattini	C₂²×D₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#1781	C₄
socle	C₂²×C₅	C₂×#342
prime radical	C₂³×D₄×D₅	C₂²
cosocle	C₂²×D₄×D₅	C₂³
hypercenter	#1781	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#7307
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #1781

5116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,5,3,4)(7,11)(8,13);(6,8)(7,10)(9,13)(11,12)(14,16,20,19)(15,18,21,17);(2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15)(16,20,21,17)(18,19)

Polycyclic presentation:

2^9,5|a^2=d,b^2=ef,e^2=i,b^a=be,c^a=cgh,c^b=cgh,d^b=dg,e^a=egi,e^b=eh,e^d=ei,f^a=fgh,f^b=fh,f^e=fi,g^a=gi,g^b=gi,h^a=hi,h^b=hi,j^a=j^3,j^d=j^4

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242830, AGL(1,𝔽₅)×#7307, #72242884
maximal normal subgroup of	#72242830, AGL(1,𝔽₅)×#7307, #72242884

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