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 | |
4470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(5,8)(13,21)(18,22)(19,24)(20,23)(26,28)(30,32)(33,36)(37,39);(2,4)(3,7)(10,15)(12,16)(18,22)(20,23)(25,26,31,28)(27,32,35,30)(29,36,38,33)(34,37,40,39);(1,2,6,4)(3,5,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,31,35)(26,32,28,30)(29,34,38,40)(33,39,36,37);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28,31,26)(27,30,35,32)(29,36,38,33)(34,37,40,39);(1,4,6,2)(3,8,7,5)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,38)(26,33,28,36)(27,34,35,40)(30,37,32,39);(1,5,6,8)(2,7,4,3)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30)(26,27)(28,35)(29,37)(31,32)(33,34)(36,40)(38,39)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017644);