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 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,15,19)(11,16,22,23)(12,14,21,17)(13,18,20,24)(25,26)(27,29)(28,30)(31,33)(32,36)(34,38)(35,39)(37,40);(2,6)(3,7)(9,11,15,22)(10,16,19,23)(12,13,21,20)(14,18,17,24)(25,27,31,36)(26,32,33,29)(28,34,37,39)(30,35,40,38);(1,3,5,7)(2,4,6,8)(9,12,15,21)(10,17,19,14)(11,20,22,13)(16,18,23,24)(25,28,31,37)(26,30,33,40)(27,34,36,39)(29,38,32,35);(1,4)(2,7)(3,6)(5,8)(9,12)(10,17)(11,13)(14,19)(15,21)(16,24)(18,23)(20,22)(25,28)(26,30)(27,34)(29,38)(31,37)(32,35)(33,40)(36,39);(9,13)(10,18)(11,21)(12,22)(14,23)(15,20)(16,17)(19,24)(25,29,31,32)(26,27,33,36)(28,35,37,38)(30,39,40,34);(9,14,15,17)(10,12,19,21)(11,18,22,24)(13,23,20,16)(25,30)(26,28)(27,35)(29,39)(31,40)(32,34)(33,37)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961597);