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 | |
4836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(3,5)(9,10)(11,15)(12,13)(14,16)(17,18)(19,28)(20,21)(22,31)(23,27)(24,30)(25,29)(26,32);(1,2,6,4)(3,8,5,7)(10,13)(14,15)(17,19,23,30)(18,24,27,28)(20,26,25,31)(21,22,29,32);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,25)(18,21,27,29)(19,26,30,31)(22,28,32,24);(1,3,6,5)(2,7,4,8)(9,11,12,16)(10,14,13,15)(17,21,23,29)(18,25,27,20)(19,22,30,32)(24,31,28,26);(1,4,6,2)(3,7,5,8)(17,18)(19,24)(20,21)(22,26)(23,27)(25,29)(28,30)(31,32);(1,5,6,3)(2,7,4,8)(17,22)(18,26)(19,29)(20,28)(21,30)(23,32)(24,25)(27,31)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066191);