Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,4)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,33,29,31,28,37)(27,34,36,39,35,32,38,40);(1,2)(3,8)(4,7)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(27,35)(32,34)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,31)(27,35)(33,37)(36,38);(1,3,5,4)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,27,29,35)(26,32,31,34)(28,38,30,36)(33,40,37,39);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28,29,30)(26,33,31,37)(27,36,35,38)(32,39,34,40)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6317799);