Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,21)(19,24,23,20)(25,26)(27,31)(28,29)(30,32);(1,3)(2,4)(5,7)(6,8)(9,10,14,12)(11,15,13,16)(18,21)(19,23)(26,28)(30,31);(1,4,5,8)(2,3,6,7)(9,11,14,13)(10,15,12,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,32)(26,30,28,31);(1,4)(2,3)(5,8)(6,7)(9,12,14,10)(11,15,13,16)(17,20,22,24)(18,23,21,19);(1,2)(3,4)(5,6)(7,8)(9,13,14,11)(10,15,12,16)(17,21,22,18)(19,24,23,20)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4862434);