Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,22)(23,24)(25,26,29,32)(27,30,28,31);(2,4)(3,7)(9,10,12,14)(11,13,16,15)(17,18,21,22)(19,20,24,23)(25,27,29,28)(26,30,32,31);(2,4)(6,8)(11,16)(13,15)(18,22)(20,23)(25,27,29,28)(26,31,32,30);(1,3,5,7)(2,6,4,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,24)(18,23,22,20)(25,28,29,27)(26,30,32,31);(1,4,5,2)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,21,23)(18,19,22,24)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038151);