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 | |
4396 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)(27,34)(28,36)(29,32)(30,31)(33,38)(35,39)(37,40);(2,5)(6,7)(9,11)(10,12)(13,21)(14,18)(15,16)(17,24)(19,22)(20,23)(25,27,31,29)(26,32,30,34)(28,35,38,37)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(9,12,15,18)(10,16,14,11)(13,20,22,24)(17,21,23,19)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,2)(3,7)(4,5)(6,8)(9,13)(10,17)(11,19)(12,20)(14,23)(15,22)(16,21)(18,24)(25,29,31,27)(26,32,30,34)(28,37,38,35)(33,39,36,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,10)(11,18,16,12)(13,17,22,23)(19,20,21,24)(25,30,31,26)(27,34,29,32)(28,36,38,33)(35,40,37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=g,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, 9687698);