Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | |
1898 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)(3,4)(5,6)(7,8)(9,10,13,15)(11,16,14,12)(18,21)(22,23)(25,26,30,32)(27,28,31,29);(1,3,7,6)(2,5,8,4)(9,11,13,14)(10,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,31)(26,28,32,29);(1,4,7,5)(2,3,8,6)(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,21)(23,24)(25,28,30,29)(26,27,32,31);(1,5,7,4)(2,3,8,6)(9,10)(11,12)(13,15)(14,16)(17,19,20,24)(18,22,21,23)(25,26)(27,28)(29,31)(30,32);(1,6,7,3)(2,5,8,4)(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,21)(22,24)(25,29,30,28)(26,31,32,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fg,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1210378);