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 | |
4455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(12,20)(15,22)(18,24)(21,23)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,2,6,4)(3,7,5,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(2,4)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,4,6,2)(3,8,5,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,5,6,3)(2,8,4,7)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064504);