Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(18,20)(19,23)(26,28)(29,32);(1,2,4,6)(3,5,8,7)(19,23)(22,24)(25,26,30,28)(27,32,31,29);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,15,16)(17,18,21,20)(19,22,23,24)(25,27,30,31)(26,29,28,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,26)(27,29,31,32);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,24,23,22)(25,29)(26,31)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=fhi,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3165174);