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 | #2404 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 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)(9,10,14,16)(11,15,17,12)(13,18,22,24)(19,23,21,20)(25,26)(27,28)(29,37)(30,34)(31,32)(33,39)(35,40)(36,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,17)(10,15,16,12)(13,21,22,19)(18,20,24,23)(25,27,31,30)(26,28,32,34)(29,35,39,38)(33,36,37,40);(1,3,4,7)(2,5,6,8)(10,16)(11,17)(18,24)(19,21)(26,32)(28,34)(33,37)(36,40);(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,21,24,19)(25,28,31,34)(26,27,32,30)(29,36,39,40)(33,35,37,38);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,21)(22,24)(25,29,31,39)(26,33,32,37)(27,35,30,38)(28,36,34,40);(9,13,14,22)(10,18,16,24)(11,19,17,21)(12,20,15,23)(25,30,31,27)(26,34,32,28)(29,38,39,35)(33,40,37,36)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650390);