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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,4)(6,7)(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,21)(22,24)(23,25)(26,31)(28,29)(30,32);(1,2,5,8)(3,6,4,7)(10,11)(14,15)(17,19,23,22)(18,24,25,27)(20,30,26,28)(21,29,31,32);(1,3,5,4)(2,6,8,7)(9,11,13,10)(12,15,16,14)(18,25)(20,26)(24,27)(28,30);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20,23,26)(18,21,25,31)(19,28,22,30)(24,32,27,29);(1,4,5,3)(2,7,8,6)(17,21)(18,26)(19,29)(20,25)(22,32)(23,31)(24,28)(27,30);(17,22,23,19)(18,24,25,27)(20,28,26,30)(21,29,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658894);