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 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,15,17)(11,20,22,16)(12,23,18,13)(14,24,21,19)(25,26,31,34)(27,32,29,39)(28,30,38,33)(35,40,37,36);(1,2,5,3)(4,6,8,7)(9,11,15,22)(10,16,17,20)(12,24,18,19)(13,21,23,14)(25,27,31,29)(26,32,34,39)(28,37,38,35)(30,36,33,40);(2,3)(6,7)(10,17)(12,18)(16,20)(19,24)(25,28,31,38)(26,33,34,30)(27,35,29,37)(32,36,39,40);(1,3,5,2)(4,7,8,6)(9,12,15,18)(10,13,17,23)(11,19,22,24)(14,20,21,16)(25,28)(26,30)(27,35)(29,37)(31,38)(32,40)(33,34)(36,39);(9,13)(10,18)(11,21)(12,17)(14,22)(15,23)(16,19)(20,24)(25,29,31,27)(26,32,34,39)(28,37,38,35)(30,40,33,36);(1,4)(2,6)(3,7)(5,8)(9,14,15,21)(10,19,17,24)(11,13,22,23)(12,20,18,16)(25,30)(26,28)(27,36)(29,40)(31,33)(32,37)(34,38)(35,39)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=hi,e^2=gh,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, 8014751);