Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1368 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | C23 | #1368 |
| 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 | |
188 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 188 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:
(1,2,4,8,5,9,14,20,6,10,15,21,16,22,26,30)(3,7,12,18,13,19,25,29,11,17,23,27,24,28,31,32)(33,34,37,43,39,44,55,63,41,46,56,45,42,62,40,61)(35,47,48,54,49,60,53,64,36,38,50,57,51,58,52,59);(1,3)(2,7)(4,12)(5,13)(6,11)(8,18)(9,19)(10,17)(14,25)(15,23)(16,24)(20,29)(21,27)(22,28)(26,31)(30,32)(33,35,38,43,40,48,59,62,42,49,60,63,56,53,57,34,41,36,47,45,55,50,64,44,39,51,58,61,37,52,54,46);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32)(33,36,38,45,40,50,59,44,42,51,60,61,56,52,57,46,41,35,47,43,55,48,64,62,39,49,58,63,37,53,54,34)Polycyclic presentation:
2^10|a^2=d,b^2=e,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,b^a=be,c^a=ce,c^b=chj,d^b=dij,d^c=dij,e^a=eg,e^d=ej,f^b=fj,f^c=fj,g^a=gi,i^a=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72231940, #72231943, #72231977, … |
| maximal normal subgroup of | #72231940, #72231943, #72231977, … |
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.5);;
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, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);