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 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,7)(9,10,14,18)(11,19,22,15)(12,23,21,16)(13,17,20,24)(26,31)(28,32)(33,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(26,31)(27,34)(33,39)(35,37);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,40,37,36);(1,2)(3,7)(4,5)(6,8)(9,10)(11,19)(12,16)(13,24)(14,18)(15,22)(17,20)(21,23)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,38)(39,40);(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,24,19,17)(25,28,30,32)(26,27,31,34)(29,36,38,40)(33,35,39,37);(1,3,4,8)(2,6,5,7)(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=hi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332704);