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 | |
4429 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)(11,12)(13,21)(14,16)(15,18)(17,22)(19,23)(20,24);(1,3,5,8)(2,6,4,7)(9,11,14,18)(10,15,16,12)(13,19,22,24)(17,23,21,20)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,2,5,4)(3,6,8,7)(10,16)(11,18)(13,22)(20,23)(26,31)(29,39)(32,40)(33,36);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,18)(13,17)(14,16)(19,23)(20,24)(21,22)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(9,12,14,15)(10,11,16,18)(13,20,22,23)(17,19,21,24)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(9,13,14,22)(10,17,16,21)(11,19,18,24)(12,20,15,23)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7908176);