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 | |
4671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,17,18,15)(13,20,22,16)(19,23,21,24)(25,26)(27,29)(28,32)(30,31)(33,37)(34,36)(35,38)(39,40);(1,2)(3,4)(5,6)(7,8)(10,12)(15,17)(16,20)(23,24)(28,35)(32,38)(34,39)(36,40);(2,6)(3,7)(10,12)(11,18)(16,20)(19,21)(26,31)(27,33)(32,38)(34,39);(3,7)(4,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(1,3)(2,4)(5,7)(6,8)(9,12,14,10)(11,15,18,17)(13,20,22,16)(19,23,21,24)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532679);