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 | #45046 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,15)(11,20,22,17)(12,18,13,23)(14,19,21,24)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,17)(12,14)(13,21)(15,20)(16,22)(18,24)(19,23)(26,28)(31,33)(32,36)(39,40);(1,3)(2,4)(5,7)(6,8)(9,12,16,13)(10,18,15,23)(11,14,22,21)(17,24,20,19)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,18,15,23)(11,21,22,14)(17,24,20,19)(25,28,30,26)(27,33,35,31)(29,32,38,36)(34,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,14,16,21)(10,19,15,24)(11,12,22,13)(17,23,20,18)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,4,5,6)(2,7,8,3)(9,15,16,10)(11,20,22,17)(12,23,13,18)(14,19,21,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8584516);