Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #13867 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C24×C4 | C24 |
| prime radical | C2×#37644 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
11982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17,18,14)(12,15)(13,16)(19,22,24,21)(20,23);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23);(2,5)(3,7)(11,18)(14,17)(19,24)(21,22);(9,12)(10,15)(11,19)(13,20)(14,21)(16,23)(17,22)(18,24)Polycyclic presentation:
2^10|a^2=e,e^2=i,b^a=bf,c^a=cg,c^b=cj,d^b=dh,e^b=ej,e^c=ei,f^a=fj,f^d=fj,g^a=gi,h^a=hj94 conjugacy classes — view the full character table.
| maximal subgroup of | #72233231, #72233305, #72233334, … |
| maximal normal subgroup of | #72233231, #72233305, #72233334, … |
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, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);