#72226413

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#518 ⋊ C₂²
Sylow	C₅² ⋊ C₂×#1904

Subgroups

	subgroup	G/H
center	C₂	#72223419
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂²×C₄	C₂²×D₅²
Fitting	C₅²×#518	C₂²
socle	C₂×C₅²	C₂×#386
prime radical	C₂×#13106298	C₂
cosocle	C₂²×C₄×C₅²	C₂⁴
hypercenter	#518	D₅²
hypocenter	C₅²	C₂×#1904
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #518

5708 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

310 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.6);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*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.9^4);;
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);