Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #485039 |
| commutator, Frattini, cosocle | #379 | C24 |
| prime radical | #6437630 | C2 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
3665 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 3665 conjugacy classes of subgroups — too many to draw. Only its 125 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 — 4 generators on 64 points:
(1,2,6,11)(3,13,18,42)(4,19,22,9)(5,10,27,38)(7,12,14,29)(8,28,32,17)(15,40,20,51)(16,41,24,53)(21,33,25,36)(23,34,43,39)(26,37,44,56)(30,49,50,46)(31,54,35,48)(45,62,47,61)(52,60,55,59)(57,63,58,64);(1,3,14,28)(2,8,29,42)(4,20,23,49,22,15,43,46)(5,24,44,48)(6,18,7,17)(9,30,34,40,19,50,39,51)(10,35,56,41)(11,32,12,13)(16,26,54,27)(21,47,52,64,25,45,55,63)(31,37,53,38)(33,57,59,61,36,58,60,62);(1,4,6,22)(2,9,11,19)(3,15,18,20)(5,25,27,21)(7,23,14,43)(8,30,32,50)(10,36,38,33)(12,39,29,34)(13,40,42,51)(16,47,24,45)(17,49,28,46)(26,55,44,52)(31,58,35,57)(37,59,56,60)(41,62,53,61)(48,63,54,64);(1,5,6,27)(2,10,11,38)(3,16,18,24)(4,21,22,25)(7,26,14,44)(8,31,32,35)(9,33,19,36)(12,37,29,56)(13,41,42,53)(15,45,20,47)(17,48,28,54)(23,52,43,55)(30,57,50,58)(34,59,39,60)(40,61,51,62)(46,63,49,64)Polycyclic presentation:
2^10|a^2=j,b^2=fhj,c^2=j,d^2=j,f^2=h,b^a=bf,c^a=ce,c^b=cej,d^a=dh,d^b=dhj,d^c=dj,e^b=eg,e^d=ej,f^a=fh,f^c=fgj,f^e=fij,g^a=gi,g^b=gij,g^f=gj,h^c=hj,i^b=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72284770, #72285153, #72285887, … |
| maximal normal subgroup of | #72284770, #72285153, #72285887, … |
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.10);;
gap> SetPower(coll, 2, F.6*F.8*F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);