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 | |
4091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,12,19,16)(14,21,23,17)(18,22,24,20)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,19)(10,16,13,12)(14,18,23,24)(17,22,21,20)(26,31)(27,32)(33,38)(34,39);(2,6)(3,7)(10,13)(12,16)(17,21)(20,22)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(3,7)(4,8)(9,12)(10,11)(13,19)(14,22)(15,16)(17,24)(18,21)(20,23)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,10)(11,16,19,12)(14,21,23,17)(18,22,24,20);(1,4)(2,3)(5,8)(6,7)(9,14,15,23)(10,17,13,21)(11,18,19,24)(12,20,16,22)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532642);