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 | |
4326 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)(4,6)(5,8)(9,10)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(26,30)(28,35)(31,36)(34,39);(1,2,4,6)(3,5,8,7)(9,11,15,13)(10,16,12,17)(14,21,22,19)(18,24,20,23)(27,33)(31,36)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,12)(16,17)(18,20)(23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662217);