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 | |
4778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(21,29)(26,30)(27,32)(28,31);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,14,15,16)(17,18,22,23)(19,24,25,20)(21,26,29,30)(27,31,32,28);(9,12,13,11)(10,14,15,16)(18,23)(20,24)(26,30)(28,31);(9,10)(11,14)(12,16)(13,15)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28);(1,2,5,4)(3,6,8,7)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27);(1,4,5,2)(3,7,8,6)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7586413);