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 | |
4534 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)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,36)(39,40);(1,2)(3,4)(5,6)(7,8)(10,11)(12,19)(16,18)(20,24)(26,31)(28,38)(32,34)(35,39);(2,6)(4,7)(13,21)(16,18)(20,24)(22,23)(26,31)(29,33)(32,34)(36,40);(2,6)(3,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27,30,37)(26,32,31,34)(28,39,38,35)(29,36,33,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,33)(27,35)(29,31)(30,38)(32,36)(34,40)(37,39);(1,4,5,7)(2,3,6,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,33)(26,28,31,38)(27,36,37,40)(32,39,34,35)Polycyclic presentation:
2^9|d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306162);