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 | |
4433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,16)(11,17,13,12)(14,22,23,18)(19,20,21,24)(25,26,31,34)(27,32,35,40)(28,38,37,30)(29,39,36,33);(1,3,5,8)(2,6,7,4)(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,22,24)(25,27)(26,32)(28,29)(30,33)(31,35)(34,40)(36,37)(38,39);(1,3)(2,4)(5,8)(6,7)(10,16)(12,17)(14,23)(19,21)(27,35)(28,37)(30,38)(32,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,28)(26,30)(27,36)(29,35)(31,37)(32,39)(33,40)(34,38);(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,29,31,36)(26,33,34,39)(27,37,35,28)(30,32,38,40);(9,14,15,23)(10,18,16,22)(11,19,13,21)(12,20,17,24)(25,30)(26,28)(27,33)(29,32)(31,38)(34,37)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8842018);