Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #15259 |
| commutator, Frattini, cosocle | C26 | C24 |
| prime radical | #6500010 | C2 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
5121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,4)(5,6)(9,10)(11,17,19,14)(12,15,21,23)(13,16)(18,24)(20,22);(1,3)(2,4)(5,8)(6,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,22,23,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14,19,17)(12,15)(13,16)(18,24,20,22)(21,23);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,24)(19,20)Polycyclic presentation:
2^10|a^2=e,b^2=ij,c^2=h,b^a=beg,c^a=cef,c^b=cfgij,d^a=dghi,d^b=dfgij,d^c=dh,e^b=ei,e^c=ej,e^d=ei,f^a=fj,f^b=fi,f^c=fj,g^a=gi,g^b=gi,g^c=gi,h^b=hij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72258599, #72258632, #72258857, … |
| maximal normal subgroup of | #72258599, #72258632, #72258857, … |
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.5);;
gap> SetPower(coll, 2, F.9*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
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);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);