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 | |
4186 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)(11,14,20,16)(12,22,23,18)(13,24,21,19)(25,26,32,36)(27,33,38,28)(29,40,37,34)(30,35,39,31);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,17,14)(12,21,23,13)(18,19,22,24)(25,27)(26,33)(28,36)(29,35)(30,34)(31,37)(32,38)(39,40);(1,2,5,4)(3,6,8,7)(10,17)(11,20)(12,23)(19,24)(25,28)(26,27)(29,39)(30,37)(31,40)(32,33)(34,35)(36,38);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,21)(13,20)(14,24)(15,23)(16,19)(17,22)(25,29,32,37)(26,34,36,40)(27,35,38,31)(28,39,33,30);(1,4,5,2)(3,7,8,6)(9,13,15,21)(10,19,17,24)(11,12,20,23)(14,18,16,22)(25,30,32,39)(26,35,36,31)(27,34,38,40)(28,37,33,29);(9,14,15,16)(10,20,17,11)(12,24,23,19)(13,22,21,18)(25,31,32,35)(26,30,36,39)(27,37,38,29)(28,40,33,34)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=ghi,f^2=i,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, 7808587);