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 | |
4502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,15,16)(11,17,12,13)(14,18,22,23)(19,24,20,21)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,3,7,6)(2,4,8,5)(10,16)(13,17)(18,23)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,7,5)(2,6,8,3)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,4,7,5)(2,3,8,6)(9,12,15,11)(10,17,16,13)(14,19,22,20)(18,21,23,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,5,7,4)(2,3,8,6)(9,13,15,17)(10,12,16,11)(14,21,22,24)(18,20,23,19)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,6,7,3)(2,4,8,5)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10318960);