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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,26,28,30)(27,31,32,29);(1,3,6,4)(2,7,8,5)(9,11,13,16)(10,14,12,15)(17,18,20,21)(19,22,24,23)(25,26,28,30)(27,29,32,31);(2,8)(5,7)(11,16)(14,15)(18,21)(22,23)(25,27)(26,29)(28,32)(30,31);(1,4,6,3)(2,5,8,7)(9,12,13,10)(11,14,16,15)(17,19,20,24)(18,22,21,23)(25,26)(27,29)(28,30)(31,32);(1,5)(2,4)(3,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,21)(22,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=f,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, 3784849);