#72226514

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#535 ⋊ C₂²
Sylow	C₅² ⋊ C₂×#3027

Subgroups

	subgroup	G/H
center	C₂	C₂×#13106387
commutator, cosocle	C₂³×C₅²	C₂⁵
Frattini	C₂³	C₂³×D₅²
Fitting	C₅²×#535	C₂²
socle	C₂×C₅²	C₂²×#120
hypercenter	#535	D₅²
hypocenter	C₅²	C₂×#3027
LCS	G ▷ C₂³×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

12116 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:

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

Polycyclic presentation:

2^8,5^2|a^2=h,e^2=h,b^a=bf,c^b=cfg,e^a=eh,e^b=eh,f^c=fh,g^a=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, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, 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);