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 | |
4259 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)(11,16)(12,13)(14,22)(15,17)(18,24)(19,21)(20,23)(25,26,30,33)(27,34,29,31)(28,36,38,32)(35,39,37,40);(1,2,5,8)(3,4,7,6)(9,11,15,12)(10,16,17,13)(14,19,24,20)(18,23,22,21)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(10,17)(11,12)(18,22)(19,20)(25,27,30,29)(26,31,33,34)(28,37,38,35)(32,40,36,39);(1,4)(2,7)(3,8)(5,6)(9,12,15,11)(10,13,17,16)(14,20,24,19)(18,21,22,23)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,12)(11,17)(14,23)(15,16)(18,19)(20,22)(21,24)(25,29,30,27)(26,31,33,34)(28,35,38,37)(32,40,36,39);(1,4)(2,7)(3,8)(5,6)(9,14,15,24)(10,18,17,22)(11,19,12,20)(13,21,16,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10004162);