Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2846 |
| commutator, Frattini, cosocle | C25 | C25 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
22640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,7)(10,14)(15,20)(16,21)(17,23);(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24);(1,2,4,6)(3,7,8,5)(12,19)(16,21)(17,23)(22,24);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,23)(21,24);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)Polycyclic presentation:
2^10|b^2=j,c^2=i,d^2=i,b^a=bj,c^a=cg,d^a=df,d^c=di,e^a=eh,e^c=ej,f^c=fi,f^e=fj,g^d=gi,h^d=hj106 conjugacy classes — view the full character table.
| maximal subgroup of | #72233431, #72233698, #72258441, … |
| maximal normal subgroup of | #72233698, #72258441, #72258445, … |
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.10);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);