Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
1212 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)(25,26,32,30,31,36,29,35)(27,37,28,34,38,33,39,40);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,31,38)(26,33,36,37)(28,29,39,32)(30,40,35,34);(1,3,5,4)(2,6,7,8)(25,28)(26,34)(27,29)(30,33)(31,39)(32,38)(35,37)(36,40);(1,4,5,3)(2,6,7,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,31,32)(26,35,36,30)(27,28,38,39)(33,40,37,34);(10,16)(14,17)(15,24)(22,23)(25,30,32,36,31,35,29,26)(27,34,28,33,38,40,39,37)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,d^b=df,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6313850);