#72226458

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#535 ⋊ C₄
Sylow	C₅² ⋊ #11004

Subgroups

	subgroup	G/H
center	C₂	#72226452
commutator	C₂²×C₄×C₅²	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#1044
Fitting	C₅²×#535	C₄
socle	C₂×C₅²	#1438
prime radical	C₅⋊D₅×#535	C₂
cosocle	C₂²×C₄×C₅⋊D₅	C₂³
hypercenter	#535	#1044
hypocenter	C₅²	#11004
LCS	G ▷ C₂²×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

4820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 26 points:

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

Polycyclic presentation:

2^8,5^2|a^2=d,d^2=h,f^2=h,b^a=bf,c^a=cfg,c^b=ce,d^b=dh,d^c=dh,e^a=eh,f^b=fh,f^c=fh,g^b=gh,i^a=i^2,i^d=i^4,j^a=i^3j^3,j^d=j^4

Conjugacy classes & character table

178 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.4);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);