Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13090 |
| commutator | C2×C42 | C23×C4 |
| Frattini, cosocle | #532 | C24 |
| prime radical | C2×#3500 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3159 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)(3,8)(4,18,23,9)(5,10,20,36)(6,12,26,11)(7,13)(14,39,42,56)(15,28,40,59)(16,30)(17,29,25,38)(19,44,47,31)(21,34,51,32)(22,45,52,61)(24,37)(27,41,57,55)(33,49,62,53)(35,50)(43,58)(46,63,48,60)(54,64);(1,3)(2,8)(4,19)(5,15)(6,25)(7,16)(9,31)(10,28)(11,29)(12,38)(13,30)(14,22)(17,26)(18,44)(20,40)(21,46)(23,47)(24,54)(27,33)(32,60)(34,63)(35,58)(36,59)(37,64)(39,45)(41,49)(42,52)(43,50)(48,51)(53,55)(56,61)(57,62);(1,4,20,49)(2,9,10,33)(3,14,40,46)(5,22,7,21)(6,23,50,52)(8,27,28,31)(11,18,35,45)(12,34,37,62)(13,32,36,61)(15,19,16,41)(17,42,54,47)(24,53,26,51)(25,55,43,48)(29,39,58,44)(30,56,59,60)(38,57,64,63);(4,21)(9,32)(14,41)(18,34)(19,46)(22,49)(23,51)(27,56)(31,60)(33,61)(39,57)(42,55)(44,63)(45,62)(47,48)(52,53)Polycyclic presentation:
2^10|a^2=g,c^2=ei,f^2=ij,h^2=j,c^a=cf,c^b=ce,d^c=dj,e^a=ei,f^a=fh,f^b=fi,g^c=ghi,g^e=gj,g^f=gij,h^a=hi,h^b=hj,h^g=hj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72286971, #72287007, #72287086, … |
| maximal normal subgroup of | #72286971, #72287007, #72287086, … |
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.7);;
gap> SetPower(coll, 3, F.5*F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);