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 | |
4692 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)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,34)(28,33)(29,38)(30,31)(32,35)(36,40)(37,39);(1,3,5,8)(2,6,7,4)(12,19)(15,17)(20,24)(22,23)(26,31)(28,38)(32,34)(36,39);(1,3)(2,4)(5,8)(6,7)(10,11)(15,17)(16,18)(22,23)(26,31)(27,35)(29,33)(36,39);(1,4)(2,3)(5,6)(7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,35)(26,32,31,34)(28,39,38,36)(29,37,33,40);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,33)(27,36)(29,31)(30,38)(32,37)(34,40)(35,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,33)(26,28,31,38)(27,37,35,40)(32,39,34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9360278);