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 | |
4368 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)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24);(1,2,5,4)(3,6,8,7)(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,22)(21,23)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=h,f^2=gh,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, 7808243);