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 | |
4394 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)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,32,38)(27,33,29,35)(28,36,30,34)(31,37,40,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27)(26,33)(28,37)(29,32)(30,39)(31,34)(35,38)(36,40);(1,3)(2,5)(4,8)(6,7)(12,21)(16,20)(19,22)(23,24)(25,28,32,30)(26,34,38,36)(27,39,29,37)(31,33,40,35);(3,8)(5,6)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,29,32,27)(26,35,38,33)(28,39,30,37)(31,34,40,36);(1,2)(3,5)(4,7)(6,8)(9,13,14,11)(10,15,17,18)(12,22,21,19)(16,23,20,24)(25,30,32,28)(26,36,38,34)(27,39,29,37)(31,33,40,35);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,20)(13,18)(14,17)(16,21)(19,24)(22,23)(25,31)(26,37)(27,34)(28,33)(29,36)(30,35)(32,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8610691);