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 | |
4617 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)(3,6)(4,5)(7,8)(9,10,14,17)(11,13,19,15)(12,16,20,23)(18,21,24,22)(25,26,30,34)(27,31,36,40)(28,37,38,32)(29,33,35,39);(2,5)(6,7)(12,20)(16,23)(18,24)(21,22)(25,27,30,36)(26,31,34,40)(28,35,38,29)(32,33,37,39);(9,11,14,19)(10,15,17,13)(12,18,20,24)(16,22,23,21)(25,28,30,38)(26,32,34,37)(27,29,36,35)(31,39,40,33);(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23)(25,28)(26,32)(27,35)(29,36)(30,38)(31,33)(34,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,12,14,20)(10,16,17,23)(11,18,19,24)(13,21,15,22)(25,29,30,35)(26,33,34,39)(27,28,36,38)(31,37,40,32);(1,3,4,8)(2,6,5,7)(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,35)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820858);