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 | |
4347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,18)(13,19)(14,16)(15,21)(20,23)(22,24)(26,31)(28,34)(32,40)(35,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(10,16)(13,24)(15,17)(18,20)(26,31)(28,34)(32,40)(35,38);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,22)(13,20)(14,21)(15,16)(18,24)(19,23)(25,27,30,37)(26,32,31,40)(28,38,34,35)(29,39,33,36);(1,3,5,4)(2,7,6,8)(9,12,14,23)(10,18,16,20)(11,19,21,22)(13,15,24,17)(25,28,30,34)(26,33,31,29)(27,35,37,38)(32,39,40,36);(1,4,5,3)(2,8,6,7)(9,13)(10,19)(11,20)(12,15)(14,24)(16,22)(17,23)(18,21)(25,29,30,33)(26,34,31,28)(27,36,37,39)(32,38,40,35)Polycyclic presentation:
2^9|c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982750);