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 | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,12,19,16)(14,17,22,21)(18,20,24,23)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(26,32)(27,28)(34,37)(35,36);(2,5)(6,7)(14,22)(17,21)(18,24)(20,23)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(3,8)(6,7)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,3,4,8)(2,6,5,7)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,17,13,21)(11,18,19,24)(12,20,16,23)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8269387);