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 | #49974 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,16)(11,12,17,15)(13,18,21,23)(19,20,24,22)(25,26,31,29)(27,34,28,32)(30,33,38,37)(35,40,36,39);(1,2,5,6)(3,7,4,8)(9,11,14,17)(10,15,16,12)(13,19,21,24)(18,22,23,20);(10,16)(11,17)(13,21)(20,22)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(9,12,14,15)(10,17,16,11)(13,22,21,20)(18,19,23,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,16,23)(11,19,17,24)(12,20,15,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982650);