Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #500441 |
| commutator, Frattini, cosocle | C22×C16 | C24 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ #33432 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,4,10,21,38,58,8,6,13,22,39,5,12,19,30,7,11,23,29,24,36,54,28,25,41,53,42,26,40,55,43)(3,14,16,45,50,62,60,44,18,9,31,33,17,46,49,59,15,35,20,56,51,34,61,63,48,32,57,37,47,64,52,27);(2,8,11,28)(4,19)(5,24)(6,25)(9,27,32,59)(10,36,29,38)(12,42,40,39)(13,43,41,30)(14,44,35,63)(16,49)(17,51)(18,48)(20,52)(21,26)(22,58)(23,55)(31,60)(33,46,37,64)(34,56,62,45)(47,50)(53,54)(57,61);(1,3)(2,9,11,32)(4,20)(5,17)(6,18)(7,15)(8,27,28,59)(10,37,29,33)(12,34,40,62)(13,35,41,14)(16,23)(19,52)(21,50)(22,57)(24,51)(25,48)(26,47)(30,44,43,63)(31,53)(36,64,38,46)(39,45,42,56)(49,55)(54,60)(58,61);(9,32)(14,35)(27,59)(33,37)(34,62)(44,63)(45,56)(46,64)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=g,f^2=hij,h^2=ij,i^2=j,b^a=bfij,c^a=cij,d^a=de,d^c=dg,e^c=ej,f^b=fh,f^c=fj,g^a=gj,h^b=hi,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72225018, C2×#72225005 |
| maximal normal subgroup of | #72225018, C2×#72225005 |
| maximal quotient of | C2×#72225005 |
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.6);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);