Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2056 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)(4,8)(5,6)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,23,24,21)(25,26)(27,28)(29,30)(31,32);(2,6)(4,7)(10,13)(11,15)(19,24)(21,23)(26,30)(28,32);(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,29,31)(26,28,30,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=g,c^2=fghi,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1959533);