Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62330 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
747 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)(3,17)(4,22,29,53)(5,30)(6,15)(7,14)(8,13)(9,16)(10,19)(11,27,44,61)(12,31)(18,49,51,41)(20,46)(21,47)(23,54,56,64)(24,48)(25,57,59,36)(26,60,28,52)(32,38)(33,45)(34,37)(35,40)(39,50,63,58)(42,62,43,55);(1,3,8,32,9,21,6,19)(2,10,15,47,16,38,13,17)(4,23,28,18,29,56,26,51)(5,7,20,35,33,34,31,24)(11,39,43,36,44,63,42,57)(12,37,45,40,46,14,30,48)(22,41,60,64,53,49,52,54)(25,55,50,27,59,62,58,61);(1,4,24,55)(2,11,40,52)(3,18,5,25)(6,26,34,61)(7,27,8,28)(9,29,35,62)(10,36,46,49)(12,41,38,57)(13,42,37,53)(14,22,15,43)(16,44,48,60)(17,39,45,64)(19,23,31,58)(20,50,32,56)(21,51,33,59)(30,54,47,63);(1,5,8,20,9,33,6,31)(2,12,15,45,16,46,13,30)(3,7,32,35,21,34,19,24)(4,25,28,50,29,59,26,58)(10,37,47,40,38,14,17,48)(11,41,43,64,44,49,42,54)(18,27,56,62,51,61,23,55)(22,39,60,36,53,63,52,57)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410699);