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 | |
4408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,20)(12,14)(13,23)(15,19)(16,24)(17,22)(18,21)(26,31)(27,34)(28,36)(37,39);(3,7)(6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,2)(3,6)(4,5)(7,8)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,3,4,7)(2,6,5,8)(9,12,15,22)(10,17,19,14)(11,21,24,13)(16,18,20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,21)(10,18,19,23)(11,22,24,12)(14,16,17,20)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(9,14,15,17)(10,12,19,22)(11,23,24,18)(13,16,21,20)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=gh,e^2=g,f^2=g,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, 8343481);