Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10336 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #111458 |
| prime radical | #6437463 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
4135 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 4135 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(1,2,5,10,7,12,23,35)(3,8,14,27,16,29,45,57)(4,17,19,47,21,48,51,64)(6,11,22,34,25,37,53,63)(9,30,32,39,33,58,61,20)(13,38,40,60,42,31,43,59)(15,36,44,62,24,28,52,56)(18,26,49,54,41,46,50,55);(1,3,7,16)(2,8,12,29)(4,18)(5,14,23,45)(6,24,25,15)(9,31)(10,27,35,57)(11,36,37,28)(13,39)(17,46)(19,50)(20,42)(21,41)(22,52,53,44)(26,48)(30,40)(32,60)(33,38)(34,62,63,56)(43,58)(47,54)(49,51)(55,64)(59,61);(1,4)(2,9)(3,13,16,42)(5,19)(6,20)(7,21)(8,26,29,46)(10,32)(11,17)(12,33)(14,43,45,40)(15,41,24,18)(22,30)(23,51)(25,39)(27,55,57,54)(28,38,36,31)(34,47)(35,61)(37,48)(44,49,52,50)(53,58)(56,59,62,60)(63,64)Polycyclic presentation:
2^10|a^2=d,b^2=hj,c^2=i,d^2=j,b^a=bf,c^a=ce,c^b=ceg,d^b=dh,d^c=di,e^a=ei,e^b=ehj,f^a=fh,f^c=fi,f^e=fj,g^a=gh,g^b=ghj,g^e=gj,h^c=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72342816, #72342852 |
| maximal normal subgroup of | #72342816, #72342852 |
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.8*F.10);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.10);;
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.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*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.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, 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);