#72226580

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×AGL(1,ℤ/8) ⋊ C₄
Sylow	C₅² ⋊ #21005

Subgroups

	subgroup	G/H
center	C₂	C₂×#13104917
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂³×#1044
Fitting	C₂×C₅²×AGL(1,ℤ/8)	C₄
socle	C₂×C₅²	C₂×#386
prime radical	C₂×AGL(1,ℤ/8)×C₅⋊D₅	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	#1044
hypocenter	C₅²	#21005
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

5284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 26 points:

(2,8)(3,12)(4,5)(7,11)(13,14)(15,16)(17,18,21,19,20)(22,23,24,26,25);(2,8)(7,15)(9,10)(11,16);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,11)(10,16);(1,2,4,9,6,8,5,10)(3,11,12,7,13,16,14,15)(18,19,20,21)(23,24,25,26)

Polycyclic presentation:

2^8,5^2|a^2=e,e^2=h,g^2=h,b^a=bg,c^a=ch,d^a=df,d^b=dg,e^b=eh,e^d=eh,f^a=fh,f^b=fh,g^b=gh,g^d=gh,i^a=i^2,i^e=i^4,j^a=i^3j^3,j^e=j^4

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

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