Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,7,5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,27)(22,25)(23,26)(24,28)(29,31)(30,32);(1,3,6,5)(2,7,4,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,30)(21,28)(22,24)(23,29)(26,32)(27,31);(2,4)(7,8)(17,20,24,23)(18,26,28,27)(19,29,22,30)(21,32,25,31);(1,4,6,2)(3,8,5,7)(17,21,24,25)(18,19,28,22)(20,31,23,32)(26,30,27,29);(17,22,24,19)(18,25,28,21)(20,29,23,30)(26,32,27,31);(1,5,6,3)(2,8,4,7)(17,23,24,20)(18,27,28,26)(19,29,22,30)(21,32,25,31)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585440);