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 | #59034 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26,32,36)(27,37,39,33)(28,40,38,31)(29,35,30,34);(1,2,5,6)(3,7,8,4)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23)(25,27)(26,33)(28,29)(30,38)(31,35)(32,39)(34,40)(36,37);(2,6)(4,7)(10,14)(11,16)(17,21)(18,23)(25,28,32,38)(26,31,36,40)(27,29,39,30)(33,35,37,34);(1,3)(2,4)(5,8)(6,7)(9,12,15,19)(10,16,14,11)(13,20,22,24)(17,23,21,18)(25,29,32,30)(26,34,36,35)(27,28,39,38)(31,33,40,37);(9,13,15,22)(10,17,14,21)(11,18,16,23)(12,20,19,24)(25,30,32,29)(26,35,36,34)(27,38,39,28)(31,37,40,33);(1,4)(2,3)(5,7)(6,8)(9,14,15,10)(11,19,16,12)(13,21,22,17)(18,24,23,20)(25,31)(26,28)(27,35)(29,33)(30,37)(32,40)(34,39)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8090098);