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 | |
4327 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)(10,11)(15,17)(16,18)(22,23)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,2,5,6)(3,7,4,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(10,11)(12,19)(16,18)(20,24)(26,33)(28,40)(32,36)(35,37);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=gh,e^2=h,f^2=g,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, 7647744);