#72242869

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#3026 ⋊ C₄
Sylow	C₅ ⋊ #364500

Subgroups

	subgroup	G/H
center	C₂	#13028194
commutator	C₂⁴×C₅	C₂³×C₄
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#3026	C₄
socle	C₂×C₅	#13967
prime radical	D₅×#3026	C₂
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	#3026	AGL(1,𝔽₅)
hypocenter	C₅	#364500
LCS	G ▷ C₂⁴×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #3026

5554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

Polycyclic presentation:

2^9,5|a^2=i,c^2=e,d^2=i,b^a=bg,c^a=cgi,c^b=cf,d^a=dh,d^c=di,e^a=ei,f^a=fi,g^c=gi,g^d=gi,h^b=hi,h^c=hi,j^c=j^3,j^e=j^4

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242889, AGL(1,𝔽₅)×#23895, #72269861, …
maximal normal subgroup of	#72242889, AGL(1,𝔽₅)×#23895, #72269861, …

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.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, 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);