Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,29,30,34,28,33)(27,35,37,40,36,32,39,38);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,36)(26,32,34,35)(28,39,31,37)(29,38,33,40);(1,3,5,4)(2,6,7,8)(27,36)(32,35)(37,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,31)(26,33,34,29)(27,37,36,39)(32,38,35,40);(10,16)(14,17)(15,24)(22,23)(25,29,31,34,30,33,28,26)(27,38,37,35,36,40,39,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6313854);