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 | #150393 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #1781 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
1061 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 1061 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2,5,10,7,12,24,35)(3,13,15,41,17,36,44,61)(4,18,20,33,22,9,30,31)(6,11,23,38,25,39,54,34)(8,26,28,45,29,53,57,16)(14,40,42,62,19,47,48,63)(21,37,51,60,52,32,50,59)(27,55,56,43,46,64,58,49);(1,3)(2,8,12,29)(4,19)(5,15)(6,16,25,45)(7,17)(9,27)(10,28,35,57)(11,36,39,13)(14,22)(18,46)(20,48)(21,49,52,43)(23,53,54,26)(24,44)(30,42)(31,58)(32,47)(33,56)(34,41,38,61)(37,40)(50,55,51,64)(59,62)(60,63);(1,4)(2,9,12,18)(3,14,17,19)(5,20,24,30)(6,21,25,52)(7,22)(8,27,29,46)(10,33)(11,37)(13,40)(15,42)(16,43)(23,50)(26,55,53,64)(28,56)(31,35)(32,39)(34,59,38,60)(36,47)(41,63,61,62)(44,48)(45,49)(51,54)(57,58)Polycyclic presentation:
2^10|a^2=d,b^2=fi,c^2=ghi,d^2=j,e^2=g,b^a=be,c^a=cfgh,c^b=cfhi,d^b=df,d^c=dhij,e^a=efg,e^b=egh,e^c=ehi,e^d=ehij,f^a=fh,f^c=fi,f^e=fij,g^a=gij,g^b=gj,g^c=gj,g^d=gj,h^b=hi,h^c=hj,h^e=hj,i^a=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72283916, #72284516, #72285153, … |
| maximal normal subgroup of | #72283916, #72284516, #72285153, … |
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.6*F.9);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
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);