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 | #51190 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,18,16,12)(13,17,22,23)(19,24,21,20)(25,26)(27,35)(28,31)(29,33)(30,39)(32,37)(34,40)(36,38);(2,6)(3,7)(9,11)(10,12)(13,21)(14,18)(15,16)(17,24)(19,22)(20,23)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(1,2,5,6)(3,4,7,8)(26,28)(32,35)(33,38)(34,39);(9,12,15,18)(10,16,14,11)(13,20,22,24)(17,21,23,19)(25,28,31,26)(27,35,37,32)(29,38,36,33)(30,39,40,34);(1,3)(2,4)(5,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,23)(15,22)(16,21)(18,24)(25,29)(26,33)(27,30)(28,38)(31,36)(32,34)(35,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,14,15,10)(11,18,16,12)(13,17,22,23)(19,20,21,24)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9703694);