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 | |
4341 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)(3,7)(4,5)(6,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,3)(2,7)(4,8)(5,6)(10,16)(11,12)(18,23)(19,20)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,4)(2,3)(5,8)(6,7)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,5,6,3)(2,4,8,7)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24);(1,4)(2,5)(3,8)(6,7)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=h,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, 8343777);