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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4665 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)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(26,27)(31,33)(32,34)(38,39);(1,2)(3,7)(4,5)(6,8)(10,15)(11,19)(12,22)(21,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,5)(6,7)(10,15)(13,17)(16,18)(21,24)(29,37)(32,34)(35,40)(38,39);(3,8)(6,7)(9,11,14,19)(10,16,15,18)(12,23,22,20)(13,21,17,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,24)(19,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,12,15,22)(11,21,19,24)(16,23,18,20)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8254648);