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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,8,4,7)(10,13)(14,16)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29);(9,10)(11,14)(12,13)(15,16)(17,19,23,24)(18,21,20,26)(22,27,30,31)(25,29,28,32);(1,3,6,4)(2,7,5,8)(19,24)(21,26)(27,31)(29,32);(1,2,6,5)(3,7,4,8)(9,11,12,15)(10,14,13,16)(17,20,23,18)(19,21,24,26)(22,25,30,28)(27,32,31,29);(1,4,6,3)(2,7,5,8)(17,21)(18,24)(19,20)(22,29)(23,26)(25,31)(27,28)(30,32);(1,5,6,2)(3,7,4,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=h,f^2=h,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, 7995145);