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 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,22)(13,17)(14,19)(15,16)(20,21)(23,24)(26,31)(28,34)(32,38)(36,40);(1,2)(3,8)(4,6)(5,7)(10,16)(12,19)(17,18)(20,23)(27,35)(32,38)(36,40)(37,39);(1,3,5,6)(2,4,7,8)(9,11,15,13)(10,17,16,18)(12,20,19,23)(14,21,22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,14)(11,20)(13,23)(15,19)(16,22)(17,21)(18,24)(25,27,30,35)(26,32,31,38)(28,36,34,40)(29,37,33,39);(9,13,15,11)(10,18,16,17)(12,23,19,20)(14,24,22,21)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(1,4)(2,6)(3,7)(5,8)(9,14)(10,19)(11,21)(12,16)(13,24)(15,22)(17,23)(18,20)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)Polycyclic presentation:
2^9|c^2=gi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332312);