Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1407 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,28,32,29,31,35,38)(27,30,34,37,33,36,39,40);(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27)(26,30)(28,34)(29,33)(31,36)(32,37)(35,39)(38,40);(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(26,31)(27,33)(32,38)(34,39);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,33)(30,36)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,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, 6294737);