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 | |
4579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(25,26,29,31)(27,30,33,39)(28,35,37,32)(34,40,36,38);(3,7)(5,8)(10,15)(12,17)(16,23)(18,21)(25,27)(26,30)(28,34)(29,33)(31,39)(32,38)(35,40)(36,37);(1,2)(3,5)(4,6)(7,8)(12,17)(13,22)(18,21)(19,24)(26,31)(27,33)(32,35)(34,36);(9,11,14,20)(10,16,15,23)(12,21,17,18)(13,19,22,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,35)(33,36)(39,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,27)(26,30)(28,36)(29,33)(31,39)(32,40)(34,37)(35,38);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31)(33,39)(34,38)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8941585);