Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #1304 |
| commutator | C24 | C43 |
| Frattini, prime radical, cosocle | C23×#41 | C23 |
| socle | C2 | C4×#1304 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C23×#131 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
2663 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 2663 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,7,12,22,34)(3,8,14,26,16,28,40,51)(4,9,18,30,20,32,45,56)(6,11,21,33,24,36,47,58)(13,25,37,48,17,29,43,54)(15,27,39,50,42,53,61,64)(19,31,44,55,23,35,46,57)(38,49,59,62,41,52,60,63);(1,3,6,15)(2,8,11,27)(4,17,19,41)(5,14,21,39)(7,16,24,42)(9,29,31,52)(10,26,33,50)(12,28,36,53)(13,23,38,20)(18,43,44,60)(22,40,47,61)(25,35,49,32)(30,54,55,63)(34,51,58,64)(37,46,59,45)(48,57,62,56);(1,4)(2,9)(3,13,16,17)(5,18)(6,23,24,19)(7,20)(8,25,28,29)(10,30)(11,35,36,31)(12,32)(14,37,40,43)(15,41)(21,46,47,44)(22,45)(26,48,51,54)(27,52)(33,57,58,55)(34,56)(38,42)(39,60)(49,53)(50,63)(59,61)(62,64)Polycyclic presentation:
2^10|a^2=d,b^2=e,c^2=g,d^2=j,c^b=cf,e^c=ei,f^b=fi,f^c=fh,g^b=gh,g^e=gj,h^b=hj,i^c=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72342789, #72342816 |
| maximal normal subgroup of | #72342789, #72342816 |
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.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);