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 | |
4579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,31,29)(27,32,28,33)(30,34,38,37)(35,39,36,40);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(26,29)(32,33)(34,37)(39,40);(1,2)(3,6)(4,5)(7,8)(10,14)(12,21)(15,17)(18,24)(25,27,31,28)(26,32,29,33)(30,36,38,35)(34,40,37,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,29,31,26)(27,32,28,33)(30,37,38,34)(35,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8941563);