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 | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,7,4,6)(11,13)(14,24)(16,20)(18,21)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(1,3,5,4)(2,6,8,7)(9,10,15,19)(11,20,13,16)(12,22,23,17)(14,18,24,21)(26,31)(27,34)(33,36)(35,40);(1,2,5,8)(3,6,4,7)(9,11,15,13)(10,16,19,20)(12,18,23,21)(14,17,24,22)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,2)(3,6)(4,7)(5,8)(9,12,15,23)(10,17,19,22)(11,21,13,18)(14,20,24,16)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,13,15,11)(10,16,19,20)(12,18,23,21)(14,22,24,17);(1,4,5,3)(2,7,8,6)(9,14)(10,18)(11,17)(12,20)(13,22)(15,24)(16,23)(19,21)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014769);