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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4245 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)(11,20)(12,23)(13,14)(15,19)(16,22)(17,24)(18,21)(26,31)(28,35)(32,34)(36,40);(1,2,4,5)(3,6,7,8)(9,11,15,22)(10,16,19,20)(12,21,24,13)(14,17,18,23)(25,26)(27,34)(28,33)(29,35)(30,31)(32,38)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(9,12,15,24)(10,17,19,23)(11,13,22,21)(14,20,18,16)(25,27)(26,32)(28,36)(29,37)(30,38)(31,34)(33,39)(35,40);(9,12)(10,17)(11,21)(13,22)(14,20)(15,24)(16,18)(19,23)(25,28,30,35)(26,33,31,29)(27,36,38,40)(32,39,34,37);(9,13,15,21)(10,18,19,14)(11,12,22,24)(16,23,20,17)(25,27)(26,34)(28,36)(29,39)(30,38)(31,32)(33,37)(35,40);(9,14,15,18)(10,13,19,21)(11,17,22,23)(12,20,24,16)(25,29,30,33)(26,35,31,28)(27,37,38,39)(32,40,34,36)Polycyclic presentation:
2^9|b^2=ghi,c^2=ghi,d^2=hi,e^2=h,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, 7861820);