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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,19)(11,20,18,14)(12,23,22,17)(13,21,24,15)(26,31)(28,32)(34,39)(36,37);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,21)(13,23)(15,22)(16,18)(17,24)(19,20)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,20)(14,24)(15,18)(16,22)(19,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,3)(2,6)(4,8)(5,7)(9,13,16,24)(10,15,19,21)(11,17,18,23)(12,20,22,14)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,14,16,20)(10,18,19,11)(12,24,22,13)(15,23,21,17);(1,2)(3,5)(4,7)(6,8)(9,15)(10,13)(11,22)(12,18)(14,23)(16,21)(17,20)(19,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329388);