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 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4377 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,18)(15,21)(19,22)(23,24)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(26,28)(27,34)(32,36)(35,40);(1,2)(3,5)(4,6)(7,8)(11,18)(14,17)(19,24)(22,23)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)Polycyclic presentation:
2^9|b^2=gh,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022754);