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 |
| prime radical | #35840 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,3,5,8)(2,6,7,4)(9,10,13,16)(11,14,19,17)(12,15,21,24)(18,22,20,23)(28,36)(31,38)(34,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965083);