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 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,21)(22,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,19,17)(13,22,24,20)(14,21,18,23)(28,36)(31,38)(34,40)(37,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,17,19,16)(13,20,24,22)(14,21,18,23)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,19,16)(13,22,24,20)(14,21,18,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(9,13)(10,18)(11,20)(12,22)(14,19)(15,24)(16,21)(17,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,14,15,18)(10,13,19,24)(11,21,12,23)(16,22,17,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8403713);