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 | #43102 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(1,3,5,8)(2,6,7,4)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10387988);