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 | #34986 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,18)(11,19,21,15)(12,22,23,16)(13,20,24,17)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,16)(14,21)(18,19)(20,23)(22,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,5)(6,8)(9,12,14,23)(10,16,18,22)(11,17,21,20)(13,19,24,15)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,20)(13,19)(14,23)(15,24)(17,21)(18,22)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,13,14,24)(10,17,18,20)(11,16,21,22)(12,19,23,15)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^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, 8013206);