Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| prime radical | C22×#2424 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
31775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(10,12)(14,16)(17,18)(19,24)(20,21)(22,23);(2,4)(5,6,8,7)(9,10)(11,12)(13,14)(15,16)(18,23)(21,24);(6,7)(9,10,11,12)(17,19,22,20)(18,21,23,24);(1,2,3,4)(5,7,8,6)(9,10)(11,12)(17,20,22,19)(18,21,23,24);(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^10|b^2=fj,c^2=i,d^2=ghi,b^a=bf,c^a=ci,c^b=ch,d^a=dh,d^b=dg,d^c=dh,e^a=efhij,e^b=efhij,e^c=efhij172 conjugacy classes — view the full character table.
| maximal subgroup of | #72333212, #72333218, #72333248, … |
| maximal normal subgroup of | #72333212, #72333218, #72333248, … |
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.6*F.10);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);