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 | |
4506 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)(17,24)(20,23)(21,22)(25,26)(27,30)(28,31)(29,33)(32,35)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,19)(12,17)(21,23)(25,27)(26,30)(28,36)(29,35)(31,40)(32,33)(34,38)(37,39);(2,5)(6,8)(10,15)(12,17)(16,18)(20,22)(25,26,29,33)(27,32,35,30)(28,31,38,37)(34,40,36,39);(2,5)(3,7)(9,11)(10,16)(12,22)(13,23)(14,19)(15,18)(17,20)(21,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,40)(33,37)(35,36);(1,2)(3,6)(4,5)(7,8)(9,12,14,17)(10,13,15,24)(11,20,19,22)(16,21,18,23)(25,26)(27,32)(28,37)(29,33)(30,35)(31,38)(34,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,21)(12,15)(14,24)(16,22)(18,20)(19,23)(25,27)(26,32)(28,36)(29,35)(30,33)(31,39)(34,38)(37,40)Polycyclic presentation:
2^9|c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8985990);