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 | #2414 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,13,16,12)(14,17,23,22)(19,21,24,20)(25,26,32,31)(27,28,34,30)(29,37,40,33)(35,39,38,36);(1,2)(3,8)(4,6)(5,7)(9,11,15,16)(10,12,18,13)(14,19,23,24)(17,20,22,21)(25,27)(26,30)(28,31)(29,38)(32,34)(33,36)(35,40)(37,39);(1,3,5,4)(2,6,7,8)(9,12,15,13)(10,11,18,16)(14,20,23,21)(17,19,22,24)(25,28,32,30)(26,27,31,34)(29,36,40,39)(33,35,37,38);(1,2)(3,6)(4,8)(5,7)(9,10)(11,12)(13,16)(14,22)(15,18)(17,23)(19,21)(20,24)(25,29,32,40)(26,33,31,37)(27,35,34,38)(28,36,30,39);(9,13,15,12)(10,16,18,11)(14,21,23,20)(17,24,22,19)(25,30,32,28)(26,34,31,27)(29,39,40,36)(33,38,37,35);(1,4,5,3)(2,6,7,8)(9,14,15,23)(10,17,18,22)(11,19,16,24)(12,20,13,21)(25,31,32,26)(27,28,34,30)(29,33,40,37)(35,39,38,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,d^2=h,e^2=g,f^2=gi,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, 8336470);