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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,18,13,17)(12,21,22,14)(19,20,23,24)(25,26,31,34)(27,29,32,28)(30,33,39,38)(35,37,40,36);(1,2)(3,6)(4,7)(5,8)(10,16)(12,22)(17,18)(19,23)(25,27,31,32)(26,28,34,29)(30,35,39,40)(33,36,38,37);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,17,16,18)(12,19,22,23)(14,20,21,24)(25,28,31,29)(26,27,34,32)(30,36,39,37)(33,35,38,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,22)(10,14,16,21)(11,19,13,23)(17,20,18,24)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,37)(36,40);(9,13,15,11)(10,18,16,17)(12,23,22,19)(14,24,21,20)(25,29,31,28)(26,32,34,27)(30,37,39,36)(33,40,38,35);(1,2)(3,5)(4,7)(6,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,22)(17,19)(18,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^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, 8329016);