Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | #14811 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2371 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)(11,12)(13,14)(15,16)(19,23)(22,24)(26,29)(30,31);(1,3,6,4)(2,7,5,8)(9,11,15,14)(10,13,16,12)(25,26)(27,30)(28,29)(31,32);(9,12,15,13)(10,11,16,14)(17,18,21,20)(19,22,23,24)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,7,5,8)(9,13,15,12)(10,11,16,14)(17,19)(18,22)(20,24)(21,23)(25,27,28,32)(26,30,29,31);(1,5,6,2)(3,8,4,7)(9,14,15,11)(10,13,16,12)(17,20,21,18)(19,24,23,22)(25,26)(27,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=i,d^2=f,e^2=fghi,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, 1293768);