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 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,15,17)(11,18,13,16)(12,14,19,24)(20,21,22,23)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(2,5)(6,7)(9,11,15,13)(10,16,17,18)(12,22,19,20)(14,21,24,23)(26,31)(27,32)(33,38)(34,39);(1,2)(3,6)(4,5)(7,8)(10,17)(12,19)(16,18)(20,22)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,20)(13,22)(15,19)(16,23)(17,24)(18,21)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,3,4,8)(2,6,5,7)(9,13,15,11)(10,18,17,16)(12,22,19,20)(14,21,24,23)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40);(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,20)(18,22)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641289);