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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,26,30,31)(27,28,32,36)(29,33,38,39)(34,35,40,37);(1,2,5,4)(3,6,8,7)(10,15)(12,17)(16,22)(20,23)(26,31)(27,32)(33,39)(34,40);(1,2,5,4)(3,6,8,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,27,30,32)(26,28,31,36)(29,34,38,40)(33,35,39,37);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28)(26,32)(27,31)(29,37)(30,36)(33,34)(35,38)(39,40);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,29,30,38)(26,33,31,39)(27,34,32,40)(28,35,36,37)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7784408);