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 |
| prime radical | C2×#538 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4356 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)(3,4)(5,6)(7,8)(9,10,15,12)(11,16,18,14)(13,17,23,20)(19,24,21,22)(25,26,30,31)(27,34,32,28)(29,33,38,39)(35,40,37,36);(2,6)(3,7)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,22,20,24)(26,31)(28,34)(33,39)(36,40);(1,3,5,7)(2,4,6,8)(11,18)(14,16)(19,21)(22,24)(25,27,30,32)(26,28,31,34)(29,37,38,35)(33,40,39,36);(1,4)(2,7)(3,6)(5,8)(9,12,15,10)(11,14,18,16)(13,17,23,20)(19,24,21,22)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,40)(38,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,28)(26,32)(27,31)(29,36)(30,34)(33,37)(35,39)(38,40);(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,21)(23,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=ghi,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, 8267034);