Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2250 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)(3,7)(4,6)(5,8)(10,14)(12,15)(17,18,20,23)(19,22,24,21)(25,26,30,29)(27,28,32,31);(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(17,19,20,24)(18,21,23,22)(25,27,30,32)(26,31,29,28);(3,8)(5,7)(10,14)(12,15)(17,18)(19,21)(20,23)(22,24)(25,28,30,31)(26,27,29,32);(1,2)(3,7)(4,6)(5,8)(9,11,13,16)(10,12,14,15)(17,18)(19,21)(20,23)(22,24)(25,29,30,26)(27,28,32,31);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23)(25,28)(26,32)(27,29)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,c^2=gi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3856108);