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 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,23)(19,21)(25,26,32,35)(27,36,31,33)(28,30,29,34)(37,39,38,40);(1,2)(3,8)(4,6)(5,7)(9,10,14,15)(11,16,18,12)(13,17,22,23)(19,24,21,20)(25,27,32,31)(26,33,35,36)(28,38,29,37)(30,39,34,40);(1,3,5,6)(2,4,7,8)(25,28,32,29)(26,34,35,30)(27,37,31,38)(33,39,36,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,22)(23,24)(25,29,32,28)(26,30,35,34)(27,38,31,37)(33,40,36,39);(9,12,14,16)(10,11,15,18)(13,20,22,24)(17,19,23,21)(25,30)(26,28)(27,39)(29,35)(31,40)(32,34)(33,38)(36,37);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,17,15,23)(11,19,18,21)(12,20,16,24)(25,31,32,27)(26,33,35,36)(28,38,29,37)(30,40,34,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8331748);