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 | |
4155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,33)(27,28,31,36)(29,32,38,40)(34,35,37,39);(1,3,5,8)(2,6,7,4)(10,14)(15,17)(16,21)(22,23)(25,27,30,31)(26,28,33,36)(29,37,38,34)(32,39,40,35);(2,7)(4,6)(12,20)(16,21)(18,24)(22,23)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,38)(39,40);(1,2,5,7)(3,6,8,4)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,37)(34,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150854);