#72227625

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#496 ⋊ C₂²
Sylow	C₅² ⋊ #61110

Subgroups

	subgroup	G/H
center	C₂	#72223052
commutator, cosocle	C₂×C₄×C₅²	C₂⁵
Frattini	C₂×C₄	C₂³×D₅²
Fitting	C₅²×#496	C₂²
socle	C₂×C₅²	C₂×D₄²
hypercenter	#496	D₅²
hypocenter	C₅²	#61110
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #496

6024 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

220 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, 3, F.7);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.6*F.8);;
gap> SetPower(coll, 6, 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.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, 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, 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);