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 | #50776 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4317 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)(10,12)(11,17)(16,20)(19,21)(26,32)(27,28)(34,39)(35,36);(1,3)(2,7)(4,8)(5,6)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(25,26)(27,33)(28,30)(29,34)(31,32)(35,40)(36,37)(38,39);(1,4)(2,3)(5,8)(6,7)(25,27,31,28)(26,33,32,30)(29,35,38,36)(34,40,39,37);(1,5,6,3)(2,4,8,7)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,28,31,27)(26,33,32,30)(29,36,38,35)(34,40,39,37);(9,12,14,10)(11,18,17,15)(13,20,22,16)(19,24,21,23)(25,29,31,38)(26,34,32,39)(27,35,28,36)(30,37,33,40);(1,4)(2,5)(3,8)(6,7)(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,30,31,33)(26,28,32,27)(29,37,38,40)(34,36,39,35)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=h,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, 8343790);