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 | #2407 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4338 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)(3,5)(4,7)(6,8)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23);(2,3)(6,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,26,30,31)(27,32,34,28)(29,33,39,40)(35,38,37,36);(1,2,5,3)(4,6,8,7)(12,21)(16,20)(18,24)(22,23)(26,31)(27,34)(33,40)(35,37);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,39)(38,40);(1,4,5,8)(2,6,3,7)(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,37,40,35);(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,29,30,39)(26,33,31,40)(27,35,34,37)(28,36,32,38)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633488);