Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #142536 |
| commutator, Frattini, cosocle | #2309 | C23 |
| prime radical | #418884 | C2 |
| LCS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1639 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 1639 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,7,12)(3,13,17,40)(4,9,22,34)(5,10,23,36)(6,11,14,27)(8,26,30,18)(15,42,44,38)(16,39,47,61)(19,49,51,64)(20,52,53,32)(21,54,25,33)(24,37,41,58)(28,57,60,45)(29,46,35,48)(31,56,62,50)(43,63,55,59);(1,3,14,26)(2,8,27,40)(4,19,21,50,22,51,25,56)(5,16,41,48)(6,18,7,17)(9,31,33,49,34,62,54,64)(10,35,58,61)(11,13,12,30)(15,43,45,53,44,55,57,20)(23,47,24,46)(28,59,42,32,60,63,38,52)(29,37,39,36);(1,4)(2,9)(3,15)(5,20)(6,25)(7,22)(8,28)(10,32)(11,33)(12,34)(13,38)(14,21)(16,19)(17,44)(18,45)(23,53)(24,55)(26,57)(27,54)(29,31)(30,60)(35,62)(36,52)(37,63)(39,49)(40,42)(41,43)(46,50)(47,51)(48,56)(58,59)(61,64)Polycyclic presentation:
2^10|a^2=j,b^2=fhj,f^2=h,b^a=bf,c^a=ce,c^b=cd,d^a=dh,d^b=dg,e^b=eghj,e^d=ej,f^a=fh,f^c=fgj,f^d=fij,f^e=fij,g^a=gi,g^b=gij,g^f=gj,h^c=hj,i^b=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72285276, #72285854, #72285887, … |
| maximal normal subgroup of | #72285276, #72285854, #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, 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.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
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, 4, F.6*F.9*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);