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 | |
4319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,7)(3,4)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(26,31)(28,32)(34,39)(36,37);(1,2)(3,5)(4,8)(6,7)(11,17)(15,19)(18,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,3,6,4)(2,5,7,8)(9,11,14,17)(10,15,13,19)(12,18,20,23)(16,22,21,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,3,6,4)(2,8,7,5)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,4,6,3)(2,8,7,5);(1,5,6,8)(2,3,7,4)(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336588);