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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(9,10,13,16)(11,15,12,14)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,28,31,32);(1,3,6,4)(2,7,8,5)(11,12)(14,15)(17,18,20,22)(19,21,24,23)(25,27,29,31)(26,28,30,32);(2,8)(5,7)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,24)(22,23)(25,28)(26,27)(29,32)(30,31);(1,4,6,3)(2,5,8,7)(9,12,13,11)(10,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,31)(30,32);(1,5)(2,4)(3,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=fghi,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, 3940193);