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 | |
4330 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)(15,21)(18,24)(22,23)(26,31)(27,32)(29,37)(36,40);(1,3,6,5)(2,7,4,8)(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(1,2,6,4)(3,8,5,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,23)(17,21)(19,24)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,38)(39,40);(1,4,6,2)(3,8,5,7)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 10362912);