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 | #2540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4091 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)(3,5)(4,7)(6,8)(9,10,15,17)(11,18,14,16)(12,13,22,23)(19,20,21,24)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(2,3)(6,7)(9,11,15,14)(10,16,17,18)(12,21,22,19)(13,20,23,24)(26,31)(27,33)(32,38)(34,39);(1,2,5,3)(4,6,8,7)(10,17)(12,22)(16,18)(19,21)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,19)(14,21)(15,22)(16,24)(17,23)(18,20)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,20)(14,24)(15,23)(16,21)(17,22)(18,19)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40);(9,14,15,11)(10,16,17,18)(12,19,22,21)(13,20,23,24)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658775);