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×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4227 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)(3,8)(4,7)(5,6)(12,20)(15,21)(19,23)(22,24)(25,26,31,34)(27,29,32,28)(30,38,39,33)(35,36,40,37);(2,6)(3,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,22)(23,24)(25,27,31,32)(26,28,34,29)(30,35,39,40)(33,36,38,37);(9,11,14,16)(10,13,17,18)(12,19,20,23)(15,22,21,24)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,40,38,35);(2,6)(4,8)(9,10)(11,13)(12,15)(14,17)(16,18)(19,22)(20,21)(23,24)(25,29,31,28)(26,27,34,32)(30,37,39,36)(33,35,38,40);(1,3,5,7)(2,4,6,8)(9,12)(10,15)(11,19)(13,22)(14,20)(16,23)(17,21)(18,24)(25,30,31,39)(26,33,34,38)(27,35,32,40)(28,36,29,37);(1,4)(2,7)(3,6)(5,8)(9,13)(10,16)(11,17)(12,22)(14,18)(15,23)(19,21)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8386739);