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 | |
4360 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)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,37)(38,40);(10,15)(11,17)(18,23)(19,24)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(1,2,5,4)(3,6,7,8)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(3,7)(6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,28,31,26)(27,32,30,34)(29,33,37,36)(35,40,38,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(1,4,5,2)(3,8,7,6)(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,30,31,27)(26,32,28,34)(29,35,37,38)(33,40,36,39)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gi,e^2=g,f^2=gh,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, 10395561);