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 | |
4623 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)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2)(3,7)(4,5)(6,8)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,5)(6,7)(10,16)(13,24)(15,19)(17,21)(29,37)(32,39)(34,40)(36,38);(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,3,4,8)(2,6,5,7)(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8121684);