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 | |
4449 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,17)(14,19)(15,16)(18,23)(20,22)(21,24);(2,3)(6,7)(9,11,15,18)(10,13,16,23)(12,20,21,14)(17,22,24,19)(25,26)(27,34)(28,29)(30,31)(32,37)(33,39)(35,36)(38,40);(1,2,5,3)(4,6,8,7)(10,16)(11,18)(14,20)(17,24)(26,31)(29,33)(32,34)(36,40);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,20)(13,22)(14,18)(15,21)(16,24)(19,23)(25,27,30,37)(26,32,31,34)(28,38,39,35)(29,36,33,40);(1,4,5,8)(2,6,3,7)(9,13,15,23)(10,18,16,11)(12,19,21,22)(14,17,20,24)(25,28)(26,33)(27,35)(29,31)(30,39)(32,36)(34,40)(37,38);(9,14,15,20)(10,19,16,22)(11,12,18,21)(13,17,23,24)(25,29,30,33)(26,28,31,39)(27,36,37,40)(32,38,34,35)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8309489);