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 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,21)(12,18)(13,24)(14,20)(15,17)(16,23)(19,22)(26,31)(29,40)(32,39)(33,37);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,19)(13,18)(14,22)(15,23)(17,21)(20,24)(25,26,30,31)(27,32,36,39)(28,37,38,33)(29,34,40,35);(10,17)(12,20)(16,21)(19,24)(26,31)(28,38)(32,39)(34,35);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,19,23,24)(13,21,22,16)(25,27,30,36)(26,32,31,39)(28,35,38,34)(29,33,40,37);(9,13,15,22)(10,19,17,24)(11,18,23,14)(12,21,20,16)(25,28)(26,33)(27,35)(29,39)(30,38)(31,37)(32,40)(34,36);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,22,23,13)(16,24,21,19)(25,29,30,40)(26,34,31,35)(27,33,36,37)(28,39,38,32)Polycyclic presentation:
2^9|b^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7623160);