Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C25 |
| commutator | C24 | C24×C4 |
| prime radical | C22×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
41777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(10,12)(13,14,15,16)(17,18,21,20)(19,22,24,23);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(18,20)(19,24);(17,19)(18,22)(20,23)(21,24);(2,4)(6,7)(17,19)(18,23)(20,22)(21,24);(1,2,3,4)(5,7,8,6)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^10|a^2=h,e^2=g,b^a=bj,c^b=cf,d^a=dhi,d^b=dhi,e^a=efg,e^b=ehi,e^d=eg160 conjugacy classes — view the full character table.
| maximal subgroup of | #72331076, #72331079, #72331247, … |
| maximal normal subgroup of | #72331076, #72331079, #72331247, … |
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, 1, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);