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 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(25,26,30,31)(27,34,32,28)(29,33,38,39)(35,40,37,36);(1,3,6,5)(2,7,4,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(26,31)(28,34)(33,39)(36,40);(1,2,6,4)(3,7,5,8)(10,15)(13,17)(16,23)(19,24)(25,27,30,32)(26,28,31,34)(29,37,38,35)(33,40,39,36);(1,3,6,5)(2,7,4,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,40)(38,39);(1,4,6,2)(3,7,5,8)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,28)(26,32)(27,31)(29,36)(30,34)(33,37)(35,39)(38,40);(1,5,6,3)(2,7,4,8)(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,29,30,38)(26,33,31,39)(27,35,32,37)(28,36,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8279357);