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×#538 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4247 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)(11,20)(12,23)(13,14)(15,16)(17,21)(18,22)(19,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,2)(3,7)(4,5)(6,8)(10,16)(11,21)(13,19)(18,23)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(2,5)(6,7)(9,11)(10,17)(12,19)(13,22)(14,23)(15,21)(16,20)(18,24)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,22)(10,18,16,23)(11,19,21,13)(14,17,24,20);(1,2)(3,6)(4,5)(7,8)(9,13,15,19)(10,14,16,24)(11,22,21,12)(17,23,20,18)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,3)(2,6)(4,8)(5,7)(9,14)(10,19)(11,23)(12,17)(13,16)(15,24)(18,21)(20,22)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8667250);