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 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(2,6)(3,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(28,36)(31,38)(34,40)(37,39);(1,3)(2,4)(5,7)(6,8)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24);(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8717518);