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 | |
4581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26)(27,29)(28,31)(30,38)(32,34)(33,40)(35,36)(37,39);(1,2,5,6)(3,7,8,4)(9,11,15,12)(10,16,17,18)(13,20,24,22)(14,21,19,23)(25,27,32,31)(26,29,34,28)(30,35,40,39)(33,37,38,36);(2,6)(4,7)(10,17)(14,19)(16,18)(21,23)(25,28,32,29)(26,27,34,31)(30,36,40,37)(33,35,38,39);(1,3)(2,4)(5,8)(6,7)(9,12,15,11)(10,18,17,16)(13,22,24,20)(14,23,19,21)(25,29,32,28)(26,27,34,31)(30,37,40,36)(33,35,38,39);(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,23)(18,21)(25,30,32,40)(26,33,34,38)(27,35,31,39)(28,36,29,37);(1,4)(2,3)(5,7)(6,8)(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,20,18,22)(25,31,32,27)(26,28,34,29)(30,39,40,35)(33,36,38,37)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8090128);