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 | #35261 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,19)(13,21)(14,15)(16,22)(18,20)(23,24)(25,26,30,33)(27,28,34,31)(29,37,38,32)(35,39,40,36);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,12,15,17)(13,18,22,24)(16,20,21,23)(27,34)(28,31)(35,40)(36,39);(1,3)(2,5)(4,8)(6,7)(10,15)(12,17)(16,21)(20,23)(25,27,30,34)(26,31,33,28)(29,35,38,40)(32,39,37,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,28)(26,27)(29,36)(30,31)(32,35)(33,34)(37,40)(38,39);(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,29,30,38)(26,32,33,37)(27,35,34,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022539);