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 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(26,31)(27,34)(33,39)(35,37);(2,6)(3,7)(10,11)(15,17)(16,18)(22,23)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,40,37,36);(1,3)(2,4)(5,7)(6,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,30,34)(26,32,31,28)(29,37,38,35)(33,36,39,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,37)(38,39);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,38)(26,33,31,39)(27,35,34,37)(28,36,32,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8324533);