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 | |
4340 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)(9,10,15,12)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,26)(27,32)(28,30)(29,39)(31,35)(33,38)(34,36)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23)(25,27,31,37)(26,32,35,40)(28,36,38,29)(30,34,33,39);(1,3,5,4)(2,6,8,7)(10,12)(16,19)(17,23)(18,24)(25,28,31,38)(26,33,35,30)(27,29,37,36)(32,34,40,39);(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,24,20,18)(25,28)(26,30)(27,36)(29,37)(31,38)(32,34)(33,35)(39,40);(1,4,5,3)(2,7,8,6)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,24,19,18)(25,29,31,36)(26,34,35,39)(27,28,37,38)(30,32,33,40);(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,23,19,17)(25,30)(26,28)(27,34)(29,40)(31,33)(32,36)(35,38)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=gi,f^2=hi,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, 7798398);