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 | |
4372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,12)(11,14,17,18)(13,20,22,16)(19,21,24,23)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,21)(22,24)(25,27,30,35)(26,28,33,31)(29,34,38,39)(32,36,40,37);(1,3,5,4)(2,6,8,7)(10,12)(11,17)(16,20)(19,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,21,24,23);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,19)(12,20)(14,23)(15,22)(17,24)(18,21)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(9,14,15,18)(10,17,12,11)(13,21,22,23)(16,19,20,24)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7790888);