Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,33,30,32,28,38)(27,35,37,39,36,29,34,40);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,36)(26,29,32,35)(28,34,31,37)(33,40,38,39);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,36)(33,38)(34,37);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,31)(26,33,32,38)(27,37,36,34)(29,39,35,40);(1,4,5,3)(2,6,7,8)(10,18)(15,19)(16,17)(23,24)(25,29,28,39,30,35,31,40)(26,34,38,36,32,37,33,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314603);