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 | |
4390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(10,11)(15,17)(16,18)(22,23)(26,31)(27,34)(29,37)(36,40);(2,5)(6,7)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(12,19)(13,21)(15,17)(16,18)(25,27,30,34)(26,32,31,28)(29,38,37,35)(33,36,39,40);(1,3,4,8)(2,6,5,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,38)(37,39);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,32)(26,27,31,34)(29,36,37,40)(33,35,39,38);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,33,31,39)(27,35,34,38)(28,36,32,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8849762);