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 | |
4439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,14,20,16)(12,22,23,18)(13,24,21,19)(25,26,31,33)(27,32,30,28)(29,37,39,34)(35,36,38,40);(1,2,4,5)(3,6,7,8)(9,11,15,20)(10,16,17,14)(12,21,23,13)(18,19,22,24)(25,27,31,30)(26,32,33,28)(29,35,39,38)(34,40,37,36);(10,17)(11,20)(12,23)(19,24)(26,33)(28,32)(34,37)(36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,20)(14,24)(15,23)(16,19)(17,22)(25,28)(26,27)(29,36)(30,33)(31,32)(34,35)(37,38)(39,40);(9,13,15,21)(10,19,17,24)(11,12,20,23)(14,18,16,22)(25,29,31,39)(26,34,33,37)(27,35,30,38)(28,36,32,40);(9,14,15,16)(10,20,17,11)(12,24,23,19)(13,22,21,18)(25,30,31,27)(26,28,33,32)(29,38,39,35)(34,36,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7868977);