Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,21)(22,24)(27,30)(28,32);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,13)(15,16)(18,20)(22,23)(25,26,29,31)(27,28,30,32);(1,3,6,4)(2,8,5,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,20)(19,22,24,23)(25,27,29,30)(26,28,31,32);(1,4,6,3)(2,8,5,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24);(1,5,6,2)(3,7,4,8)(9,12,13,11)(10,14,15,16)(17,20,21,18)(19,22,24,23)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1347431);