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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,6,4,7)(10,13)(11,15);(1,2)(3,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,23)(21,30)(24,31)(28,32);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28);(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32);(1,4,5,3)(2,7,8,6)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gh,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7559989);