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 | |
4915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,16,19,21)(12,17,22,24)(13,18,23,20)(26,30)(28,36)(31,33)(34,37);(1,3,5,8)(2,6,7,4)(10,15)(11,19)(17,24)(18,20)(26,30)(31,33)(32,38)(39,40);(1,3)(2,4)(5,8)(6,7)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(9,12)(10,17)(11,20)(13,16)(14,22)(15,24)(18,19)(21,23)(25,27,29,35)(26,31,30,33)(28,37,36,34)(32,40,38,39);(9,13)(10,18)(11,17)(12,21)(14,23)(15,20)(16,22)(19,24)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10367912);