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 | |
4730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,19)(11,18,22,16)(12,14,13,17)(20,21,23,24)(25,26,30,33)(27,28,35,31)(29,37,39,32)(34,40,38,36);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,23)(13,20)(14,21)(15,22)(17,24)(18,19)(25,26,30,33)(27,28,35,31)(29,32,39,37)(34,36,38,40);(1,3,5,8)(2,6,4,7)(9,12,15,13)(10,17,19,14)(11,20,22,23)(16,21,18,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,40)(36,37)(38,39);(1,4,5,2)(3,7,8,6)(9,13,15,12)(10,14,19,17)(11,20,22,23)(16,21,18,24)(25,28,30,31)(26,27,33,35)(29,36,39,40)(32,34,37,38);(9,11)(10,18)(12,20)(13,23)(14,21)(15,22)(16,19)(17,24)(25,27)(26,28)(29,38)(30,35)(31,33)(32,40)(34,39)(36,37);(9,14)(10,12)(11,21)(13,19)(15,17)(16,23)(18,20)(22,24)(25,29)(26,32)(27,34)(28,36)(30,39)(31,40)(33,37)(35,38)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388031);