Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
53793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(5,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,21,23,24);(10,13)(11,16)(17,19)(18,21)(20,23)(22,24);(10,13)(11,16)(18,22)(19,23);(2,4)(6,8)(11,16)(14,15)(18,22)(21,24);(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11,12,16)(10,14,13,15)(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^10|a^2=h,f^2=g,b^a=bg,c^a=ch,c^b=cgh,d^a=di,e^a=eh,e^b=eg,e^c=eh,e^d=ei,f^a=fg,f^b=fg,f^c=fh,f^d=fij,f^e=fg154 conjugacy classes — view the full character table.
| maximal subgroup of | #72232064, C2×#72232052, #72328461, … |
| maximal normal subgroup of | #72232064, C2×#72232052, #72328461, … |
| maximal quotient of | #72323745, C2×#72232052, #72331477, … |
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, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);