Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C32 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C42 |
| commutator | C4 | C8×C32 |
| Frattini, cosocle | C42×C16 | C22 |
| socle, prime radical | C23 | #1374 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 2 generators on 56 points:
(9,10)(11,17,18,23)(12,16)(13,15)(14,19,21,24)(20,22)(25,26,27,31,28,32,35,41,29,33,36,42,38,44,47,51,30,34,37,43,39,45,48,52,40,46,49,53,50,54,55,56);(1,2,3,5,4,6,7,8)(9,11)(10,14,15,21)(12,19)(13,18)(16,17,22,23)(20,24)Polycyclic presentation:
2^10|a^2=c,b^2=d,c^2=f,d^2=g,e^2=h,f^2=i,i^2=j,b^a=be,c^b=ch,d^a=dh352 conjugacy classes — view the full character table.
| maximal subgroup of | #72234289 |
| maximal normal subgroup of | #72234289 |
| maximal quotient of | #72234283, #72234284, #72234312, … |
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.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);