Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1436 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)(15,21)(19,24)(20,23)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(3,8)(5,7)(9,11,14,21)(10,15,17,18)(12,20,13,19)(16,24,22,23);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(27,33)(30,36)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314848);