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 | |
4184 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)(3,4)(5,6)(7,8)(9,10,15,18)(11,16,13,12)(14,17,23,22)(19,24,21,20)(27,33)(31,34)(35,39)(38,40);(2,6)(3,7)(9,11,15,13)(10,12,18,16)(14,19,23,21)(17,20,22,24)(25,26,30,29)(27,34,33,31)(28,36,37,32)(35,38,39,40);(1,3,5,7)(2,4,6,8)(9,12,15,16)(10,11,18,13)(14,20,23,24)(17,19,22,21)(25,27,30,33)(26,31,29,34)(28,35,37,39)(32,38,36,40);(1,4)(2,7)(3,6)(5,8)(9,10)(11,12)(13,16)(14,22)(15,18)(17,23)(19,24)(20,21)(25,28,30,37)(26,32,29,36)(27,35,33,39)(31,38,34,40);(9,13,15,11)(10,16,18,12)(14,21,23,19)(17,24,22,20);(9,14,15,23)(10,17,18,22)(11,19,13,21)(12,20,16,24)(25,29,30,26)(27,34,33,31)(28,32,37,36)(35,38,39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,e^2=ghi,f^2=gi,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, 8370408);