#72226443

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₅

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#1891

Subgroups

	subgroup	G/H
center	C₄	#13104917
commutator	C₂³×C₅²	C₂³×C₄
Frattini	C₂³	C₂³×#1044
Fitting	C₅²×#525	C₄
socle	C₂×C₅²	C₂×#386
prime radical	C₅⋊D₅×#525	C₂
cosocle	C₂²×#1039	C₂⁴
hypercenter	#525	#1044
hypocenter	C₅²	C₂×#1891
LCS	G ▷ C₂³×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #525

5620 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

232 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.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 6, 1, 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, 5, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);