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 |
| prime radical | #2540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4323 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)(6,8)(9,10,15,17)(11,16,12,14)(13,21,22,18)(19,24,20,23)(26,32)(28,38)(33,35)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,23,21,24)(25,26,31,32)(27,35,30,33)(28,34,38,29)(36,40,39,37);(10,17)(11,12)(13,22)(23,24)(25,27,31,30)(26,33,32,35)(28,36,38,39)(29,40,34,37);(9,12,15,11)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,28)(26,34)(27,36)(29,32)(30,39)(31,38)(33,37)(35,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,19)(12,20)(14,24)(15,22)(16,23)(17,21)(25,29,31,34)(26,28,32,38)(27,37,30,40)(33,39,35,36);(9,14,15,16)(10,11,17,12)(13,23,22,24)(18,20,21,19)(25,30,31,27)(26,35,32,33)(28,39,38,36)(29,37,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893315);