Finite Group DatabasebetaFinite Group Databasebeta| order | 2560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C5×#2021 ⋊ C4 |
| Sylow | C5 ⋊ #376180 |
4600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,5,3,4)(12,17)(19,20);(6,9)(7,12)(8,15)(10,18)(11,19)(13,17)(14,20)(16,21);(2,3,4,5)(6,8)(7,11,13,14)(9,15)(10,16)(12,20,17,19)(18,21);(6,7,10,13)(8,14,16,11)(9,17,18,12)(15,20,21,19)Polycyclic presentation:
2^9,5|a^2=i,b^2=e,b^a=bf,c^a=cfi,c^b=ch,d^a=dg,d^c=dh,e^a=ei,f^b=fi,g^c=gi,h^a=hi,j^b=j^3,j^e=j^4106 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#23996, #72269861 |
| maximal normal subgroup of | AGL(1,𝔽5)×#23996, #72269861 |
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, 2, 5]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);