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 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,31,29)(27,35,36,32)(28,34,30,33)(37,39,38,40);(2,5)(6,7)(10,15)(13,16)(17,20)(21,23)(26,29)(27,36)(33,34)(37,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(17,20)(19,24)(25,27,31,36)(26,32,29,35)(28,37,30,38)(33,39,34,40);(1,3,4,8)(2,6,5,7)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,28,31,30)(26,33,29,34)(27,37,36,38)(32,39,35,40);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,29,31,26)(27,35,36,32)(28,34,30,33)(37,40,38,39);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,30,31,28)(26,34,29,33)(27,38,36,37)(32,40,35,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664601);