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 | #6437628 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
4169 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 4169 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:
(1,2,5,10,7,12,22,33)(3,8,14,25,16,27,38,51)(4,9,18,29,20,31,44,56)(6,11,21,32,23,34,47,58)(13,24,35,48,36,49,59,63)(15,39,37,60,40,26,52,50)(17,28,41,53,42,54,61,64)(19,45,43,62,46,30,57,55);(1,3)(2,8)(4,17)(5,14)(6,15)(7,16)(9,28)(10,25)(11,26)(12,27)(13,19)(18,41)(20,42)(21,37)(22,38)(23,40)(24,30)(29,53)(31,54)(32,50)(33,51)(34,39)(35,43)(36,46)(44,61)(45,49)(47,52)(48,55)(56,64)(57,59)(58,60)(62,63);(1,4,7,20)(2,9,12,31)(3,13,16,36)(5,18,22,44)(6,19,23,46)(8,24,27,49)(10,29,33,56)(11,30,34,45)(14,35,38,59)(15,17,40,42)(21,43,47,57)(25,48,51,63)(26,28,39,54)(32,55,58,62)(37,41,52,61)(50,53,60,64)Polycyclic presentation:
2^10|a^2=d,c^2=j,d^2=j,b^a=bf,c^a=cg,c^b=ce,e^a=ei,f^c=fh,f^e=fj,g^b=ghi,g^e=gj,h^b=hj,i^b=ij,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72342829, #72342845, #72342859 |
| maximal normal subgroup of | #72342829, #72342845, #72342859 |
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, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);