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 | |
4682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,14)(15,18)(16,20)(22,24)(25,26)(27,28)(29,37)(30,31)(32,39)(33,34)(35,38)(36,40);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,35)(30,33)(31,34)(32,36)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(11,17)(15,18)(19,23)(22,24)(26,31)(27,33)(32,37)(35,40);(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,28)(26,27)(29,38)(30,34)(31,33)(32,40)(35,37)(36,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,26)(27,34)(28,33)(29,32)(30,31)(35,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,32)(27,35)(28,36)(30,39)(31,37)(33,40)(34,38)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8977926);