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 | |
4319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,12)(13,21)(14,15)(16,17)(18,22)(19,23)(20,24)(26,28)(31,34)(32,36)(38,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(18,21)(19,24)(27,33)(31,34)(35,39)(38,40);(2,5)(6,8)(10,15)(12,16)(18,21)(20,23)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8030189);