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 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,16)(11,13,12,17)(14,22,23,18)(19,24,20,21)(26,30)(28,35)(31,36)(34,39);(1,2,5,4)(3,7,8,6)(9,11)(10,13)(12,15)(14,19)(16,17)(18,21)(20,23)(22,24)(27,33)(31,36)(34,39)(38,40);(1,3,5,8)(2,6,4,7)(10,16)(11,12)(18,22)(19,20)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,16,13)(14,20,23,19)(18,24,22,21)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,13,15,17)(10,11,16,12)(14,21,23,24)(18,19,22,20)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,14,15,23)(10,18,16,22)(11,19,12,20)(13,21,17,24)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=gh,e^2=h,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, 8022688);