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 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,20)(12,22)(13,17)(14,21)(15,16)(18,23)(19,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2,7,8)(3,4,6,5)(10,16)(11,13)(18,22)(19,21)(26,30)(31,33)(32,37)(38,39);(1,3,7,6)(2,5,8,4)(9,11,15,13)(10,17,16,20)(12,19,23,21)(14,18,24,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,4,7,5)(2,3,8,6)(9,12,15,23)(10,18,16,22)(11,21,13,19)(14,20,24,17)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,5,7,4)(2,3,8,6)(9,13,15,11)(10,17,16,20)(12,19,23,21)(14,22,24,18)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(1,6,7,3)(2,5,8,4)(9,14)(10,19)(11,18)(12,20)(13,22)(15,24)(16,21)(17,23)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9685664);