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 | #37764 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,5,4)(10,14)(15,16)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29);(1,3,6,5)(2,7,8,4)(9,10,13,14)(11,15,12,16)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30);(1,4,6,7)(2,5,8,3)(18,24)(19,20)(26,32)(27,28);(1,4)(2,3)(5,8)(6,7)(9,11,13,12)(10,15,14,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29);(1,5,6,3)(2,4,8,7)(9,12,13,11)(10,16,14,15)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27);(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606121);