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 | |
4307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23);(1,3,6,5)(2,7,4,8)(10,15)(12,17)(16,21)(20,23)(25,26,30,35)(27,33,36,31)(28,32,38,39)(29,34,40,37);(1,3,6,5)(2,8,4,7)(10,15)(12,17)(13,22)(18,24)(25,27,30,36)(26,31,35,33)(28,34,38,37)(29,39,40,32);(9,11,14,19)(10,12,15,17)(13,18,22,24)(16,20,21,23)(25,28)(26,32)(27,34)(29,31)(30,38)(33,40)(35,39)(36,37);(1,4,6,2)(3,7,5,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,27)(26,33)(28,37)(29,32)(30,36)(31,35)(34,38)(39,40);(1,5,6,3)(2,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,23)(19,24)(25,29)(26,34)(27,32)(28,31)(30,40)(33,38)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8853430);