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, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(25,26,31,33,30,28,29,37)(27,34,36,39,35,32,38,40);(1,2,4,5)(3,6,7,8)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,27,30,35)(26,32,28,34)(29,38,31,36)(33,40,37,39);(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24)(27,35)(32,34)(36,38)(39,40);(2,5)(6,8)(10,16)(14,23)(15,19)(22,24)(25,28,31,37,30,26,29,33)(27,34,36,39,35,32,38,40);(1,3,4,7)(2,6,5,8)(25,29,30,31)(26,33,28,37)(27,36,35,38)(32,39,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6332222);