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 | |
4503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(3,7)(9,10)(11,13)(12,20)(14,15)(16,22)(17,18)(19,24)(21,23)(25,26)(27,36)(28,29)(30,39)(31,35)(32,37)(33,38)(34,40);(1,2,5,4)(3,8,7,6)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,27,31,37)(26,32,35,36)(28,34,38,39)(29,30,33,40);(1,2,5,4)(3,8,7,6)(10,15)(11,17)(16,20)(19,23)(25,28,31,38)(26,29,35,33)(27,34,37,39)(30,36,40,32);(1,3,5,7)(2,6,4,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,22)(21,23)(25,28)(26,33)(27,34)(29,35)(30,36)(31,38)(32,40)(37,39);(1,2,5,4)(3,6,7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24)(25,29,31,33)(26,28,35,38)(27,30,37,40)(32,34,36,39);(1,4,5,2)(3,8,7,6)(9,13,14,18)(10,17,15,11)(12,21,22,24)(16,23,20,19)(25,30)(26,34)(27,33)(28,36)(29,37)(31,40)(32,38)(35,39)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014547);