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, socle | C2 | #111636 |
| commutator | C2×#131 | C22×C4 |
| Frattini, cosocle | C22×#131 | C23 |
| prime radical | #6629044 | C2 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
2875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(3,14,17,23)(4,21)(5,26)(6,7)(9,31,33,35)(10,22)(11,27)(12,30)(15,39,44,57)(16,25)(18,50,49,62)(19,42,20,41)(28,29)(32,37)(34,55,46,63)(36,59)(38,61)(40,52,58,51)(45,53,64,47)(48,54);(1,2)(3,15,17,44)(4,22)(5,27,28,61)(6,12)(7,30)(8,13)(9,18)(10,21)(11,26,38,29)(14,39,23,57)(16,37)(19,47,20,53)(24,59,60,36)(25,32)(31,62)(33,49)(34,51)(35,50)(40,63)(41,64,42,45)(43,54,56,48)(46,52)(55,58);(1,3,16,42,6,19,25,14,8,17,21,41,7,20,4,23)(2,9,32,63,30,46,37,31,13,33,22,55,12,34,10,35)(5,18,48,62,26,51,60,40,28,49,54,50,29,52,24,58)(11,15,43,39,61,53,59,45,38,44,56,57,27,47,36,64)Polycyclic presentation:
2^10|a^2=d,b^2=dg,c^2=deij,e^2=i,f^2=ij,h^2=j,i^2=j,b^a=bg,c^a=ce,c^b=cf,d^c=dj,e^a=ei,e^b=ei,e^c=ej,f^a=fh,f^b=fh,g^c=ghi,g^f=gj,h^a=hj,i^a=ij,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72282517, #72341802, #72341888 |
| maximal normal subgroup of | #72282517 |
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.4*F.7);;
gap> SetPower(coll, 3, F.4*F.5*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, 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);