Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139504 |
| commutator | C2×#117 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | #417800 | C2 |
| LCS | G ▷ C2×#117 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35616 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1733 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1733 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,2,7,12)(3,8,16,28)(4,17)(5,10,23,34)(6,11,24,35)(9,29)(13,36)(14,40,42,26)(15,27,44,57)(18,45)(19,31,50,60)(20,46)(21,33)(22,47)(25,43)(30,48)(32,58)(37,62,63,53)(38,54)(39,56)(41,59)(49,61)(51,64)(52,55);(1,3)(2,8)(4,18)(5,14,23,42)(6,15,24,44)(7,16)(9,30)(10,26,34,40)(11,27,35,57)(12,28)(13,20)(17,45,47,59)(19,37)(21,52,29,48)(22,41)(25,32)(31,53)(33,55)(36,46,56,64)(38,49,43,58)(39,51)(50,63)(54,61)(60,62);(1,4,19,49)(2,9,31,46)(3,13,37,48,16,39,63,52)(5,20,6,21)(7,22,50,58)(8,25,53,59,28,54,62,45)(10,32,11,17)(12,33,60,64)(14,41,44,38,42,18,15,43)(23,51,24,29)(26,55,57,36,40,30,27,56)(34,61,35,47)Polycyclic presentation:
2^10|a^2=ij,b^2=f,c^2=defi,d^2=gij,e^2=j,b^a=bhi,c^a=ce,c^b=cd,d^a=dg,d^b=dgh,d^c=dghj,e^b=egij,e^c=egij,f^a=fi,f^c=fhij,f^d=fij,f^e=fj,g^b=gj,h^b=hi,h^c=hij,h^d=hj,i^c=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72310943, #72311052, #72340436 |
| maximal normal subgroup of | #72310943, #72311052, #72340436 |
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.9*F.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.9);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);