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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,22)(23,24)(26,30)(31,37)(32,38)(39,40);(2,6)(4,7)(9,11,15,13)(10,16,20,17)(12,23,19,21)(14,22,18,24)(26,30)(27,33)(32,38)(34,35);(1,3)(2,7)(4,6)(5,8)(9,11,15,13)(10,17,20,16)(12,21,19,23)(14,22,18,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,4,5,7)(2,3,6,8)(9,12,15,19)(10,18,20,14)(11,21,13,23)(16,22,17,24)(25,27,29,33)(26,31,30,37)(28,35,36,34)(32,40,38,39);(9,13,15,11)(10,16,20,17)(12,23,19,21)(14,24,18,22);(1,3)(2,4)(5,8)(6,7)(9,14,15,18)(10,19,20,12)(11,22,13,24)(16,21,17,23)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8362892);