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 | |
4593 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)(9,10,15,18)(11,16,19,24)(12,22,21,14)(13,23,20,17)(25,26)(27,32)(28,29)(30,38)(31,33)(34,35)(36,39)(37,40);(2,6)(3,7)(9,11)(10,16)(12,13)(14,17)(15,19)(18,24)(20,21)(22,23)(25,27)(26,32)(28,36)(29,39)(30,34)(31,37)(33,40)(35,38);(1,3,5,7)(2,4,6,8)(11,19)(12,21)(14,22)(16,24)(26,33)(27,37)(28,35)(30,39);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,20)(13,19)(15,21)(16,23)(17,24)(18,22)(25,28)(26,34)(27,38)(29,33)(30,40)(31,35)(32,39)(36,37);(9,13,15,20)(10,17,18,23)(11,21,19,12)(14,16,22,24)(25,29,31,34)(26,35,33,28)(27,39,37,30)(32,38,40,36);(9,14)(10,12)(11,17)(13,16)(15,22)(18,21)(19,23)(20,24)(25,30,31,39)(26,36,33,38)(27,34,37,29)(28,40,35,32)Polycyclic presentation:
2^9|a^2=ghi,c^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, 8842020);