Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152477 |
| commutator | #492 | C22×C4 |
| Frattini, prime radical, cosocle | #3151 | C23 |
| LCS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,8)(4,18)(5,23,26,41)(6,11)(7,13)(9,33,34,59)(10,38,40,27)(12,28)(14,43)(15,47,24,39)(16,32)(17,42)(19,51)(20,37,25,35)(21,52)(22,53)(29,55,60,54)(30,58,57,49)(31,50)(36,56)(44,63)(45,61,48,62)(46,64);(1,3)(2,8)(4,19)(5,24,26,15)(6,17)(7,16)(9,29,34,60)(10,39,40,47)(11,31)(12,42)(13,32)(14,21)(18,51)(20,55,25,54)(22,46)(23,57,41,30)(27,58,38,49)(28,50)(33,48,59,45)(35,61,37,62)(36,43)(44,56)(52,63)(53,64);(1,4,7,22)(2,9,12,35,13,34,11,37)(3,14)(5,25,26,20)(6,21,28,56)(8,29,42,61,32,60,31,62)(10,36,23,53,40,52,41,18)(15,48)(16,44)(17,46)(19,50)(24,45)(27,59,38,33)(30,51,39,43,57,64,47,63)(49,54)(55,58)Polycyclic presentation:
2^10|a^2=d,b^2=d,c^2=gj,f^2=hj,g^2=i,h^2=j,b^a=be,c^a=cdfh,c^b=cghi,d^c=di,e^c=eg,f^a=fhij,f^b=fgi,f^c=fij,f^d=fj,f^e=fh,g^a=gh,g^b=gij,g^e=gi,h^a=hj,h^b=hi,h^e=hj,i^a=ij37 conjugacy classes — view the full character table.
| maximal subgroup of | #72245425 |
| maximal normal subgroup of | #72245425 |
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.4);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);