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 | |
4367 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)(3,7)(4,5)(6,8)(9,10,16,15)(11,12,18,14)(13,21,24,17)(19,23,22,20)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(2,5)(6,7)(9,11)(10,14)(12,15)(13,22)(16,18)(17,20)(19,24)(21,23)(27,33)(31,35)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,14)(10,11,15,18)(13,20,24,23)(17,19,21,22)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(1,3,4,8)(2,6,5,7)(9,13,16,24)(10,17,15,21)(11,19,18,22)(12,20,14,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,14,16,12)(10,18,15,11)(13,23,24,20)(17,22,21,19);(9,15,16,10)(11,12,18,14)(13,17,24,21)(19,23,22,20)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40)Polycyclic presentation:
2^9|a^2=gi,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336589);