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 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,21)(13,14)(15,20)(16,22)(18,23)(19,24)(26,31)(32,33)(34,37)(39,40);(2,5)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,23)(19,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,24)(17,23)(18,20)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(1,3,4,8)(2,6,5,7)(9,13,14,10)(11,17,20,15)(12,21,22,16)(18,24,19,23)(25,28,30,27)(26,33,31,32)(29,35,38,36)(34,39,37,40);(9,11)(10,17)(12,18)(13,15)(14,20)(16,24)(19,22)(21,23)(25,29,30,38)(26,34,31,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=g,f^2=gh,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, 8570125);