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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2256 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)(18,20)(22,23)(25,26,30,32)(27,31,29,28);(1,2,6,5)(3,8,4,7)(10,12)(15,16)(17,18,21,20)(19,23,24,22)(25,27,30,29)(26,28,32,31);(1,3,6,4)(2,7,5,8)(9,11,14,13)(10,15,12,16)(17,18,21,20)(19,22,24,23)(25,28,30,31)(26,27,32,29);(1,4,6,3)(2,8,5,7)(9,12,14,10)(11,16,13,15)(17,19,21,24)(18,22,20,23)(25,26)(27,28)(29,31)(30,32);(1,5,6,2)(3,8,4,7)(9,13,14,11)(10,16,12,15)(17,20,21,18)(19,23,24,22)(25,29,30,27)(26,31,32,28)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=fh,d^2=g,e^2=fh,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, 762423);