Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C26 |
| commutator | C23 | C25×C4 |
| prime radical | C24×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
81389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(11,15)(13,16)(17,18,20,22)(19,23,24,21);(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24);(2,3)(6,8)(9,10,12,14)(11,13,15,16);(1,3,4,2)(5,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24);(17,19)(18,21)(20,24)(22,23);(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^10|a^2=h,c^2=j,d^2=ghi,b^a=bg,c^a=ci,c^b=cj,d^a=dgi,d^c=dij,e^a=eg,e^d=eg,f^a=fgj,f^d=fgj208 conjugacy classes — view the full character table.
| maximal subgroup of | #72339320, C2×#72339319, #72339895, … |
| maximal normal subgroup of | #72339320, C2×#72339319, #72339895, … |
| maximal quotient of | #72339325, C2×#72339319, #72339933 |
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, 3, F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);