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 | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,31,28,30,32,29,34)(27,35,37,36,38,33,40,39);(3,8)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(26,32)(28,34)(33,35)(36,39);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27)(26,33)(28,39)(29,40)(30,38)(31,37)(32,35)(34,36);(2,6)(5,7)(11,20)(15,18)(19,21)(23,24)(25,28,31,32,30,34,29,26)(27,36,37,33,38,39,40,35);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,29,30,31)(26,34,32,28)(27,37,38,40)(33,39,35,36)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6331457);