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 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(10,15)(11,17)(18,23)(19,21)(25,26,31,32)(27,36,37,33)(28,40,34,29)(30,35,38,39);(3,7)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(26,32)(27,37)(28,34)(35,39);(1,2,5,4)(3,6,7,8)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,16,15,12)(13,21,22,19)(18,20,23,24)(25,28,31,34)(26,29,32,40)(27,38,37,30)(33,39,36,35);(1,4,5,2)(3,8,7,6)(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29)(26,34)(27,39)(28,32)(30,33)(31,40)(35,37)(36,38);(9,13,14,22)(10,18,15,23)(11,19,17,21)(12,20,16,24)(25,30,31,38)(26,35,32,39)(27,34,37,28)(29,33,40,36)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=g,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, 8332750);