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 | #34880 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1491 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,28,32,29,33,36,40)(27,30,35,37,34,31,38,39);(2,5)(3,7)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(10,15)(11,17)(16,22)(18,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,38)(39,40);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(2,5)(6,8)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,34)(30,31)(35,38)(37,39)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6317154);