Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14,17,19)(15,18,20,16)(21,22,27,30)(23,24,25,28)(26,29,35,36)(31,32,33,34);(13,15,17,20)(14,18,19,16)(21,23,27,25)(22,28,30,24)(26,31,35,33)(29,34,36,32);(6,9)(10,12)(14,19)(15,20)(21,24,27,28)(22,23,30,25)(26,34,35,32)(29,33,36,31);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,25,27,23)(22,24,30,28)(26,33,35,31)(29,32,36,34);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,19,15)(21,26,27,35)(22,29,30,36)(23,31,25,33)(24,32,28,34)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928601);