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×#555 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4545 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)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32);(1,3,7,5)(2,4,8,6)(9,11,12,16)(10,13,14,15)(19,25)(23,27)(26,31)(30,32);(1,4,7,6)(2,5,8,3)(17,19,22,25)(18,23,21,27)(20,26,28,31)(24,30,29,32);(1,3,7,5)(2,4,8,6);(1,5,7,3)(2,4,8,6)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,6,7,4)(2,5,8,3)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7790216);