Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 8369 conjugacy classes of subgroups; only the 1049 normal subgroups are shown.
| 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)(14,16)(17,18)(19,21)(20,22)(23,24);(5,6)(7,8)(10,11)(13,14)(15,16)(18,19)(20,24);(2,4)(5,6,8,7)(10,11)(18,19)(22,23);(2,4)(6,7)(9,10,12,11)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(5,7,8,6)(9,11,12,10)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^10|c^2=fh,d^2=fi,e^2=i,b^a=bj,c^a=cf,c^b=cfh,d^a=df,d^b=dhi,d^c=dhi,e^a=eg,e^b=ehi,e^c=egi,e^d=eg124 conjugacy classes — view the full character table.
| maximal subgroup of | #72331120, #72331129, #72331142, … |
| maximal normal subgroup of | #72331120, #72331129, #72331142, … |
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, 3, F.6*F.8);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);