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) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,3)(2,7)(4,5)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2,6,4)(3,7,8,5)(10,14)(15,17)(16,20)(22,23)(26,28)(31,33)(32,36)(38,39);(2,4)(3,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,6,2)(3,7,8,5)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,5,6,7)(2,3,4,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227591);