Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator, socle, prime radical | C23 | C23×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,27,32,28,23,33,36,26,31,42,48,25,35,29,43)(24,30,40,47,38,34,45,50,39,46,49,52,41,37,44,51);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,27,36,28,31,33,48,26,35,42,43,25,22,29,32)(24,37,40,51,38,30,45,47,39,34,49,50,41,46,44,52);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,26,39)(22,30,31,46)(23,34,35,37)(25,41,28,38)(27,40,42,49)(29,44,33,45)(32,47,48,52)(36,50,43,51);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,25,26,28)(22,23,31,35)(24,38,39,41)(27,29,42,33)(30,37,46,34)(32,36,48,43)(40,45,49,44)(47,51,52,50)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404283);