Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1333 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | #2215 | C8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
322 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 512 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,8)(4,16)(5,10)(6,11)(7,12)(9,18)(13,24)(14,22)(15,23)(17,25)(19,32)(20,28)(21,29)(26,30)(27,31);(1,3,5,4,7,15,21,19,6,14,20,18,13,30,17,31)(2,8,10,16,12,23,29,32,11,22,28,9,24,26,25,27)(34,39,41,51,42,52,57,44)(35,40,45,55)(36,47)(37,50)(43,59,58,53,48,54,62,64)(46,61,60,56)(49,63);(1,4,17,3,13,31,20,30,6,18,21,14,7,19,5,15)(2,9,25,22,24,32,28,23,11,16,29,8,12,27,10,26)(33,34,40,54,63,58,35,44,50,57,61,64,49,43,60,39,38,42,55,59,47,62,45,51,37,41,56,53,36,48,46,52)Polycyclic presentation:
2^10|b^2=d,c^2=defg,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,c^a=ch,c^b=ce,d^c=dij,e^b=eg,e^c=ei,e^d=ej,f^c=fj,g^b=gi,g^c=gj,i^b=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72287159, #72287247, #72287265, … |
| maximal normal subgroup of | #72287159, #72287247, #72287265, … |
| maximal quotient of | #72287169, #72323176 |
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, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);