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 | |
4754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,16)(11,15,20,17)(12,18,23,22)(13,21,24,19)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(3,7)(5,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,17)(18,24)(19,23)(25,26,29,32)(27,30,35,33)(28,31,37,40)(34,38,36,39);(1,2)(3,5)(4,6)(7,8)(10,16)(11,20)(12,23)(13,24)(25,27,29,35)(26,30,32,33)(28,36,37,34)(31,39,40,38);(9,11)(10,17)(12,19)(13,22)(14,20)(15,16)(18,24)(21,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,21)(13,17)(14,23)(15,24)(16,22)(19,20)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,19)(11,22)(12,17)(14,24)(15,23)(16,21)(18,20)(25,28)(26,31)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948284);