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 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,40,37);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27)(26,31)(28,37)(29,39)(30,36)(32,40)(33,38)(34,35);(1,3,5,4)(2,7,6,8)(10,17)(12,20)(16,19)(21,23)(27,36)(28,38)(31,35)(32,39);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,28,30,38)(26,32,34,39)(27,37,36,33)(29,31,40,35);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,35,38,31)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894483);