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 | #2682 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10,12,14)(11,13,15,16)(18,23)(20,30)(24,26)(27,31);(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,26,29,24)(20,25,30,21)(27,28,31,32);(1,3,6,4)(2,7,5,8)(18,23)(20,30)(24,26)(27,31);(1,2,6,5)(3,7,4,8)(17,19)(18,24)(20,31)(21,32)(22,29)(23,26)(25,28)(27,30);(1,4,6,3)(2,7,5,8)(17,20)(18,25)(19,27)(21,23)(22,30)(24,32)(26,28)(29,31);(1,5,6,2)(3,7,4,8)(9,11,12,15)(10,13,14,16)(17,21,22,25)(18,20,23,30)(19,28,29,32)(24,27,26,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995158);