Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C25 |
| commutator | C24 | C24×C4 |
| prime radical | C22×#2433 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
40809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(13,14)(15,16)(17,18)(19,20);(1,2,4,3)(6,8)(10,11)(14,16)(17,18,19,20);(1,2)(3,4)(17,18)(19,20);(2,3)(5,6)(7,8)(9,10,12,11)(13,14)(15,16)(17,18,19,20);(1,3,4,2)(5,6)(7,8)(9,11,12,10)(13,14)(15,16)(17,18)(19,20)Polycyclic presentation:
2^10|b^2=h,d^2=g,e^2=gh,b^a=bf,c^b=ch,d^a=dhi,d^b=dfj,d^c=dh,e^b=efhj,e^c=ei,e^d=eh196 conjugacy classes — view the full character table.
| maximal subgroup of | #72298589, #72298617, #72336080, … |
| maximal normal subgroup of | #72298589, #72298617, #72336080, … |
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, 2]);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);