Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #13654 |
| commutator | C32 | C22×C8 |
| Frattini, cosocle | C4×C32 | C23 |
| prime radical | C2×D64 | C4 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ C8×C16 ◁ #7329 ◁ G | |
| LECS | G ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,4,6,5,7,8,3)(10,16,19,35)(11,22)(12,27)(13,28)(14,31)(17,26)(18,33,42,38)(20,47,45,65)(21,49,46,62)(23,50)(24,30)(25,57)(29,56)(32,60)(34,63,48,36)(37,68,66,44)(39,64,67,61)(40,54)(41,53)(43,69,70,71)(51,52)(55,72)(58,59);(9,10,17,39,12,20,25,37,28,44,52,47,59,61,11,16,14,18,22,36,29,43,51,49,32,46,57,71,24,34,26,38,15,19,40,67,30,45,53,66,60,68,55,65,56,64,23,35,31,42,50,63,58,70,72,62,13,21,41,69,27,48,54,33);(1,3,8,7,5,6,4,2)(10,18,19,42)(11,23)(16,33,35,38)(17,40)(20,43,45,70)(21,44,46,68)(22,50)(25,53)(26,54)(34,64,48,61)(36,39,63,67)(37,62,66,49)(41,57)(47,69,65,71)(51,72)(52,55)Polycyclic presentation:
2^10|a^2=e,b^2=dfij,c^2=eg,d^2=fhj,e^2=g,f^2=hi,h^2=ij,i^2=j,b^a=bd,c^b=cij,d^a=df,d^c=dj,e^b=ej,f^a=fh,h^a=hi,i^a=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72230706, #72230779, #72230785 |
| maximal normal subgroup of | #72230706, #72230779, #72230785 |
| maximal quotient of | #72230779 |
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.5);;
gap> SetPower(coll, 2, F.4*F.6*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.7);;
gap> SetPower(coll, 4, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);