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 | |
4406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(25,26)(27,37)(28,34)(29,33)(30,39)(31,36)(32,38)(35,40);(3,7)(5,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,31,38)(26,32,36,37)(28,35,29,39)(30,33,40,34);(1,2)(3,5)(4,6)(7,8)(10,14)(12,21)(15,17)(18,22)(25,28,31,29)(26,33,36,34)(27,35,38,39)(30,32,40,37);(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,29,31,28)(26,34,36,33)(27,35,38,39)(30,32,40,37);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,30)(26,35)(27,33)(28,37)(29,32)(31,40)(34,38)(36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=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, 8947850);