Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×#120 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C27 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
36079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,23,24,22);(3,7)(5,8)(12,21)(15,20)(18,24)(22,23);(12,21)(15,20)(18,24)(22,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,3,4,7)(2,5,6,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(2,6)(5,8)(11,19)(14,17)(18,24)(22,23)Polycyclic presentation:
2^10|a^2=h,e^2=ij,d^a=dg,e^a=ehj,e^b=ei,e^c=ej,e^d=eij,f^d=fi,g^a=gj,g^e=gj,h^d=hj136 conjugacy classes — view the full character table.
| maximal subgroup of | #72232684, #72234665, #72234791, … |
| maximal normal subgroup of | #72232684, #72234665, #72234791, … |
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.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);