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 | #42995 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,19,13,16)(12,14,20,18)(21,22,23,24)(26,31)(27,35)(33,40)(34,37);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,24,18,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,39)(35,38)(36,40);(1,3,4,7)(2,6,5,8)(10,17)(11,13)(12,20)(22,24)(26,31)(28,36)(32,38)(34,37);(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,22,19,24)(25,27,30,35)(26,32,31,38)(28,34,36,37)(29,33,39,40);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,22,18,24)(25,28)(26,33)(27,34)(29,38)(30,36)(31,40)(32,39)(35,37);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,21,19,23)(25,29,30,39)(26,34,31,37)(27,33,35,40)(28,32,36,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gi,d^2=i,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894761);