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 | |
4449 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)(11,17)(12,15)(13,21)(14,16)(18,22)(19,23)(20,24)(29,37)(33,38)(34,39)(36,40);(1,2,5,6)(3,7,4,8)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24)(25,26,30,31)(27,28,32,35)(29,33,37,38)(34,36,39,40);(10,16)(11,12)(18,21)(19,20)(26,31)(28,35)(33,38)(36,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,27,30,32)(26,28,31,35)(29,34,37,39)(33,36,38,40);(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|b^2=ghi,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617997);