Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187195 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#538 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,12,28,10,36,38)(3,14,17,46,16,45,26,58)(4,20,23,53,22,41,55,63)(6,9,27,35,7,13,29,44)(8,30,34,15,32,49,43,48)(11,39,42,25,37,57,61,24)(18,47,51,64,19,40,52,62)(21,31,54,59,50,33,56,60);(1,3,6,18)(2,8,9,31)(4,21,24,48)(5,26,27,52)(7,19,28,16)(10,32,13,33)(11,40,20,45)(12,43,35,60)(14,37,47,41)(15,22,50,25)(17,29,51,36)(23,56,39,49)(30,55,54,57)(34,44,59,38)(42,64,53,46)(58,61,62,63);(2,9)(4,22)(5,27)(8,31)(10,13)(11,41)(14,40)(15,48)(17,51)(20,37)(21,50)(23,57)(24,25)(26,52)(29,36)(30,56)(32,33)(39,55)(42,61)(45,47)(46,58)(49,54)(53,63)(62,64);(2,10)(8,32)(9,13)(11,37)(12,38)(14,45)(20,41)(31,33)(34,43)(35,44)(40,47)(42,61)(46,58)(53,63)(59,60)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94569);