Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #446242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15836 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
813 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)(3,15,20,42)(4,21)(5,11,30,40)(6,12)(7,13)(8,14)(9,18,33,19)(10,25)(16,44,47,31)(17,38)(22,51,46,34)(23,53,56,50)(24,37)(26,36)(27,57)(28,39,29,41)(32,48)(35,54,61,55)(43,49)(45,63)(52,62)(58,60)(59,64);(1,3,7,19,8,20,6,18)(2,9,12,15,14,33,13,42)(4,22,25,16,26,46,24,47)(5,27,29,17,30,48,28,49)(10,34,21,44,37,51,36,31)(11,38,39,57,40,43,41,32)(23,45,55,52,56,64,54,58)(35,59,53,62,61,63,50,60);(1,4,7,25,8,26,6,24)(2,10,12,21,14,37,13,36)(3,16,18,22,20,47,19,46)(5,23,28,54,30,56,29,55)(9,31,42,51,33,44,15,34)(11,35,41,50,40,61,39,53)(17,45,48,52,49,64,27,58)(32,59,38,62,57,63,43,60);(1,5)(2,11,14,40)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32,33,57)(10,35,37,61)(12,41,13,39)(15,43,42,38)(16,45)(18,48)(19,27)(20,49)(21,50,36,53)(22,52)(24,55)(25,54)(26,56)(31,59,44,63)(34,60,51,62)(46,58)(47,64)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353438);