#72226522

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#535 ⋊ C₂²
Sylow	C₅² ⋊ C₂×#3021

Subgroups

	subgroup	G/H
center	C₂	C₂×#13106387
commutator, cosocle	C₂×C₄×C₅²	C₂⁵
Frattini	C₂×C₄	C₂³×D₅²
Fitting	C₅²×#535	C₂²
socle	C₂×C₅²	C₂²×#120
hypercenter	#535	D₅²
hypocenter	C₅²	C₂×#3021
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

11764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Polycyclic presentation:

2^8,5^2|a^2=h,g^2=h,c^a=cg,c^b=cf,e^b=eh,f^a=fh,g^a=gh,g^b=gh,g^c=gh,i^a=i^4,i^d=i^4,j^a=i^3j,j^d=j^4

Conjugacy classes & character table

262 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.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, 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.9^4);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);