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 | #42504 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,4)(10,13)(11,12)(18,21)(19,20)(25,26,30,34)(27,31,32,28)(29,37,38,33)(35,36,40,39);(1,2,5,6)(3,7,4,8)(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,2,5,6)(3,7,4,8)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,2)(3,7)(4,8)(5,6)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014481);