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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4111 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)(11,13)(12,17)(14,23)(15,16)(18,20)(19,22)(21,24)(25,26,32,37)(27,28,34,33)(29,38,39,35)(30,31,36,40);(2,5)(6,7)(9,11)(10,13)(12,20)(14,21)(15,19)(16,22)(17,18)(23,24)(25,27,32,34)(26,33,37,28)(29,31,39,40)(30,35,36,38);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(12,21)(18,23)(25,28,32,33)(26,34,37,27)(29,30,39,36)(31,35,40,38);(9,12,15,21)(10,17,16,24)(11,20,19,14)(13,18,22,23)(25,29,32,39)(26,35,37,38)(27,31,34,40)(28,30,33,36);(1,2)(3,7)(4,5)(6,8)(9,13)(10,11)(12,18)(14,24)(15,22)(16,19)(17,20)(21,23)(25,30)(26,31)(27,38)(28,39)(29,33)(32,36)(34,35)(37,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,20)(10,18,16,23)(11,21,19,12)(13,17,22,24)(25,31)(26,36)(27,39)(28,35)(29,34)(30,37)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9688239);