Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10980 |
| commutator | C24×C4 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| prime radical | #686216 | C2 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#2339 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
4600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,7)(10,14)(12,19)(15,23)(16,22)(17,20)(21,24);(1,2)(3,7,8,5)(4,6)(10,14,16,22)(15,21,24,23);(1,3)(2,5)(4,8)(6,7)(9,10,11,15,13,16,18,24)(12,14,20,21,19,22,17,23)Polycyclic presentation:
2^10|b^2=g,c^2=d,d^2=j,e^2=j,b^a=bg,c^a=ce,c^b=cef,d^a=dh,e^a=ej,e^b=ei,e^c=ehj,e^d=ej,f^c=fh,f^d=fj,g^c=gij,h^c=hj70 conjugacy classes — view the full character table.
| maximal subgroup of | #72260380, C2×#72238546, #72260773, … |
| maximal normal subgroup of | #72260380, C2×#72238546, #72260773, … |
| maximal quotient of | C2×#72238546 |
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, 2, F.7);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);