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 | |
4309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,13,17)(11,15,19,14)(12,20,21,16)(18,22,24,23)(26,31)(28,32)(33,39)(36,37);(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(1,3,6,5)(2,7,4,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(1,3,6,5)(2,8,4,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,4,6,2)(3,8,5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,37)(36,40)(38,39);(1,5,6,3)(2,8,4,7)(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358856);