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 | |
4896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,23)(21,29)(22,26)(25,31)(27,28)(30,32);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(17,19,23,27)(18,24,20,28)(21,26,31,30)(22,25,32,29);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(18,20)(19,27)(25,29)(26,30);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,20,23,18)(19,24,27,28)(21,29,31,25)(22,26,32,30);(1,3)(2,5)(4,7)(6,8)(17,21)(18,25)(19,26)(20,29)(22,28)(23,31)(24,32)(27,30);(17,22,23,32)(18,26,20,30)(19,25,27,29)(21,24,31,28)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gi,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, 8657533);