Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,7,5)(9,10,13,14)(11,12,16,15)(17,18,22,21)(19,20,24,23)(26,28)(30,31);(1,3)(2,5)(4,8)(6,7)(10,14)(12,15)(19,24)(20,23)(25,26,29,28)(27,31,32,30);(1,2,6,4)(3,5,7,8)(9,11)(10,15)(12,14)(13,16)(17,19,22,24)(18,23,21,20)(25,27,29,32)(26,30,28,31);(1,4,6,2)(3,8,7,5)(9,12,13,15)(10,11,14,16)(17,20)(18,19)(21,24)(22,23)(25,26,29,28)(27,30,32,31);(1,5,6,8)(2,7,4,3)(9,11)(10,15)(12,14)(13,16)(17,21,22,18)(19,20,24,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 758360);