Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
2274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(6,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(10,11)(12,19)(16,18)(20,24)(27,33)(30,37)(34,39)(38,40);(1,2,4,3)(5,6)(7,8)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(1,3,4,2)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4304788);