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×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4514 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)(3,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(18,23)(21,25)(24,30)(27,31);(1,3,6,5)(2,4,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,24)(20,21)(22,23)(25,29)(26,27)(28,30)(31,32);(1,3,6,5)(2,7,8,4)(18,23)(20,29)(24,30)(26,32);(1,4,6,7)(2,3,8,5)(9,12,13,10)(11,15,16,14)(17,19,22,28)(18,24,23,30)(20,26,29,32)(21,27,25,31);(1,3,6,5)(2,4,8,7)(17,20)(18,25)(19,26)(21,23)(22,29)(24,31)(27,30)(28,32);(1,5,6,3)(2,7,8,4)(17,21,22,25)(18,20,23,29)(19,27,28,31)(24,26,30,32)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=hi,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, 7588548);