Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510223 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31396 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,11,6,13,30,23)(3,14,19,39,20,9,17,41)(4,21,25,5,26,29,10,27)(8,33,32,62,31,12,45,44)(15,50,52,37,51,57,59,48)(16,22,53,42,54,61,55,38)(18,40,34,46,58,49,56,64)(24,47,35,36,63,60,43,28);(1,3)(2,9,13,14)(4,22)(5,16,27,54)(6,20)(7,17)(8,34)(10,42)(11,39,23,41)(12,46,33,64)(15,24)(18,32)(19,30)(21,53,29,55)(25,38)(26,61)(28,37,36,48)(31,56)(35,59)(40,62,49,44)(43,52)(45,58)(47,50,60,57)(51,63);(1,4,7,10,6,26,30,25)(2,5,23,29,13,27,11,21)(3,15,17,52,20,51,19,59)(8,35,32,24,31,43,45,63)(9,37,41,50,14,48,39,57)(12,47,62,36,33,60,44,28)(16,49,55,64,54,40,53,46)(18,42,56,61,58,38,34,22);(1,5,6,27)(2,10,13,25)(3,16,20,54)(4,23,26,11)(7,29,30,21)(8,36,31,28)(9,38,14,42)(12,35,33,43)(15,49,51,40)(17,55,19,53)(18,57,58,50)(22,39,61,41)(24,44,63,62)(32,60,45,47)(34,48,56,37)(46,52,64,59)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=ehi,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417422);