Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15924 |
| 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 | |
811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,27,8,12,7,13)(3,9,29,32,16,34,17,33)(4,18,21,10,23,48,22,37)(5,11,25,40,26,38,28,39)(14,41,50,58,43,35,19,30)(15,44,46,60,24,31,45,59)(20,51,53,36,54,47,55,62)(42,63,49,57,56,61,52,64);(1,3,7,29,8,16,6,17)(2,9,27,33,12,34,13,32)(4,19,22,14,23,50,21,43)(5,24,28,45,26,15,25,46)(10,35,48,58,37,41,18,30)(11,31,40,60,38,44,39,59)(20,52,55,42,54,49,53,56)(36,61,47,57,62,63,51,64);(1,4)(2,10,12,37)(3,14,16,43)(5,20)(6,21)(7,22)(8,23)(9,30)(11,36,38,62)(13,18,27,48)(15,42,24,56)(17,19,29,50)(25,53)(26,54)(28,55)(31,57)(32,41)(33,35)(34,58)(39,51,40,47)(44,64)(45,52,46,49)(59,61)(60,63);(1,5,7,25,8,26,6,28)(2,11,27,39,12,38,13,40)(3,15,29,45,16,24,17,46)(4,20,22,53,23,54,21,55)(9,31,33,59,34,44,32,60)(10,36,48,51,37,62,18,47)(14,42,50,52,43,56,19,49)(30,57,35,61,58,64,41,63)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,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, 356090);