Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478918 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17175 |
| 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 | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,14)(3,15,18,35,19,9,32,43)(4,20,24,10,23,38,25,39)(5,26,29,11,30,42,31,41)(16,46,49,33,48,36,21,44)(17,40,51,34,52,60,27,45)(22,55,57,53,28,37,56,62)(47,59,58,61,50,64,54,63);(1,3,6,18,8,19,7,32)(2,9,14,43,12,15,13,35)(4,21,25,16,23,49,24,48)(5,27,29,52,30,51,31,17)(10,36,20,44,39,46,38,33)(11,40,42,45,41,60,26,34)(22,50,57,58,28,47,56,54)(37,59,62,63,55,64,53,61);(1,4,6,25,8,23,7,24)(2,10,14,20,12,39,13,38)(3,16,18,49,19,48,32,21)(5,28,29,56,30,22,31,57)(9,33,43,36,15,44,35,46)(11,37,42,62,41,55,26,53)(17,50,27,58,52,47,51,54)(34,59,40,63,45,64,60,61);(1,5,8,30)(2,11,12,41)(3,17,19,52)(4,22,23,28)(6,31,7,29)(9,34,15,45)(10,37,39,55)(13,42,14,26)(16,47,48,50)(18,51,32,27)(20,53,38,62)(21,54,49,58)(24,57,25,56)(33,59,44,64)(35,40,43,60)(36,61,46,63)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386115);