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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,7)(3,4)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2,6,7)(3,5,4,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(26,32)(27,28)(34,37)(35,36);(1,3,6,4)(2,5,7,8)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,4,6,3)(2,5,7,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,5)(2,4)(3,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8370388);