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 | |
4508 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)(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,24,22,20)(26,31)(27,32)(33,38)(34,39);(2,5)(6,7)(9,11,15,13)(10,16,17,18)(12,22,23,19)(14,20,21,24)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,2)(3,6)(4,5)(7,8)(10,17)(14,21)(16,18)(20,24)(29,37)(33,38)(34,39)(35,40);(1,3,4,8)(2,6,5,7)(9,12)(10,14)(11,19)(13,22)(15,23)(16,24)(17,21)(18,20)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(9,13,15,11)(10,18,17,16)(12,19,23,22)(14,24,21,20)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,14)(10,12)(11,20)(13,24)(15,21)(16,22)(17,23)(18,19)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202727);