#72226392

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×AGL(1,ℤ/8) ⋊ C₄
Sylow	C₅² ⋊ #21249

Subgroups

	subgroup	G/H
center	C₂	C₂×#13104918
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂³×#1044
Fitting	C₂×C₅²×AGL(1,ℤ/8)	C₄
socle	C₂×C₅²	C₂×#390
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	#21249
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

4844 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:

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

Polycyclic presentation:

2^8,5^2|a^2=e,b^2=h,f^2=h,g^2=h,b^a=bg,c^a=cgh,c^b=cf,d^b=dh,f^b=fh,f^c=fh,g^a=gh,g^b=gh,g^c=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, 2, F.8);;
gap> SetPower(coll, 6, 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.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, 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);