Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,13,14)(11,15,16,12)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,3,6,5)(2,7,4,8)(9,10)(11,12)(13,14)(15,16)(17,18,21,20)(19,23,24,22)(27,34)(30,33)(35,40)(38,39);(2,4)(7,8)(10,14)(11,16)(17,19,21,24)(18,22,20,23)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,4,6,2)(3,8,5,7)(9,11,13,16)(10,12,14,15)(25,27,29,34)(26,30,32,33)(28,35,37,40)(31,38,36,39);(1,5,6,3)(2,7,4,8)(9,12)(10,11)(13,15)(14,16)(17,20,21,18)(19,23,24,22)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fghi,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2685092);