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 | |
4408 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)(3,5,7,8)(25,26,32,30)(27,31,35,33)(28,34,29,38)(36,39,37,40);(3,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,24)(20,23)(21,22)(25,27)(26,33)(28,36)(29,37)(30,31)(32,35)(34,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,22)(21,24)(25,28,32,29)(26,34,30,38)(27,36,35,37)(31,39,33,40);(9,11)(10,15)(12,20)(13,19)(14,17)(16,23)(18,22)(21,24)(25,29,32,28)(26,34,30,38)(27,37,35,36)(31,39,33,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,21)(13,17)(15,19)(16,22)(18,24)(20,23)(25,30,32,26)(27,31,35,33)(28,38,29,34)(36,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,21)(19,20)(25,31)(26,35)(27,30)(28,39)(29,40)(32,33)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9117690);