Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,8)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(26,31)(27,34)(29,37)(36,40);(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,26,30,31)(27,32,34,28)(29,33,37,39)(35,40,38,36);(5,6)(7,8)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27)(26,32)(28,31)(29,38)(30,34)(33,36)(35,37)(39,40);(1,2,3,4)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,30,32)(26,27,31,34)(29,36,37,40)(33,35,39,38);(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,29,30,37)(26,33,31,39)(27,35,34,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4323978);