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 | |
4472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26,31,34)(27,28,29,32)(30,33,38,40)(35,36,37,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27,31,29)(26,32,34,28)(30,37,38,35)(33,36,40,39);(10,16)(15,17)(18,23)(22,24)(25,28,31,32)(26,27,34,29)(30,36,38,39)(33,35,40,37);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,29,31,27)(26,28,34,32)(30,35,38,37)(33,39,40,36);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,30,31,38)(26,33,34,40)(27,35,29,37)(28,36,32,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,e^2=g,f^2=g,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, 7881164);