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 | |
4460 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)(11,17)(12,20)(15,18)(16,21)(25,26)(27,30)(28,29)(31,33)(32,35)(34,39)(36,37)(38,40);(3,7)(5,8)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27,31,35)(26,32,33,30)(28,38,34,36)(29,37,39,40);(1,2)(3,5)(4,6)(7,8)(10,14)(11,17)(16,21)(19,22)(26,33)(28,34)(30,32)(36,38);(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,28)(26,29)(27,36)(30,37)(31,34)(32,40)(33,39)(35,38);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,36)(35,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30,31,32)(26,35,33,27)(28,37,34,40)(29,38,39,36)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840479);