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 | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,19,20,16)(12,22,23,14)(13,24,21,18)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,17,19)(12,21,23,13)(14,18,22,24);(1,2,5,4)(3,6,8,7)(10,17)(11,20)(12,23)(18,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,21)(13,20)(15,23)(16,18)(17,22)(19,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(9,13,15,21)(10,18,17,24)(11,12,20,23)(14,16,22,19)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,4,5,2)(3,7,8,6)(9,14)(10,12)(11,18)(13,19)(15,22)(16,21)(17,23)(20,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7808571);