Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,4,5,7)(10,13)(11,15)(17,18)(19,24)(20,23)(21,22)(25,26)(27,30)(28,29)(31,32);(1,3,6,5)(2,7,8,4)(10,13)(14,16)(17,18,22,21)(19,23,20,24)(26,29)(30,31);(2,8)(3,5)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,21,24)(25,26)(27,31)(28,29)(30,32);(1,4,6,7)(2,3,8,5)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,21,24);(1,5,6,3)(2,7,8,4)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,23,20,24)(25,27,28,32)(26,30,29,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=gi,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4626839);