Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10336 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #116351 |
| prime radical | #62811 | C22 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
2909 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 2909 conjugacy classes of subgroups — too many to draw. Only its 155 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)(3,13,15,8)(4,18,20,50)(6,11,24,36)(7,12,25,37)(9,30,32,21)(14,42,43,31)(16,40,47,28)(17,41,48,29)(19,51,52,27)(22,34,23,55)(26,38,57,62)(33,56,35,54)(39,53,58,45)(44,61,46,60)(49,63,64,59);(1,3,6,16,7,17,26,49)(2,8,11,28,12,29,38,59)(4,19,21,14,22,45,54,44)(5,15,24,47,25,48,57,64)(9,31,34,58,33,60,18,27)(10,13,36,40,37,41,62,63)(20,53,30,46,23,52,56,43)(32,61,55,51,35,42,50,39);(1,4,7,22)(2,9,12,33)(3,14,17,44)(5,23)(6,21,26,54)(8,27,29,58)(10,35)(11,34,38,18)(13,39)(15,46)(16,45,49,19)(20,25)(24,56)(28,31,59,60)(30,57)(32,37)(36,50)(40,61)(41,51)(42,63)(43,48)(47,52)(53,64)(55,62)Polycyclic presentation:
2^10|a^2=d,b^2=f,c^2=ghi,f^2=j,b^a=bde,c^a=cefij,c^b=cfg,d^b=dh,d^c=dh,e^a=eh,e^b=eh,e^c=ei,g^a=gi,g^b=gj,g^c=gj,g^d=gj,h^c=hj,i^a=ij,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72317506, #72317672 |
| maximal normal subgroup of | #72317506, #72317672 |
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);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
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, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);