#72269865

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#2883 ⋊ C₄
Sylow	C₅ ⋊ #355922

Subgroups

	subgroup	G/H
center	C₂	#13027984
commutator	C₂³×C₅	C₂²×C₄²
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#2883	C₄
socle	C₂×C₅	#13794
prime radical	C₂×D₅×H(5,𝔽₂)	C₂²
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	#2883	AGL(1,𝔽₅)
hypocenter	C₅	#355922
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #2883

5410 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)(6,9)(7,12)(8,16)(10,18)(11,19)(13,17)(14,20)(15,21);(6,8)(7,11)(9,16)(10,15)(12,19)(13,14)(17,20)(18,21);(2,3,4,5)(8,15)(16,21);(6,7)(8,14,15,11)(9,17,18,12)(10,13)(16,19)(20,21)

Polycyclic presentation:

2^9,5|a^2=f,b^2=e,b^a=bfg,c^a=ch,c^b=chi,d^a=dh,f^c=fi,f^d=fi,g^d=gi,h^a=hi,j^b=j^3,j^e=j^4

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269866, AGL(1,𝔽₅)×#21776, #72269870
maximal normal subgroup of	#72269866, AGL(1,𝔽₅)×#21776, #72269870

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