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 | |
4383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,31,30)(27,32,29,34)(28,33,37,36)(35,39,38,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,29)(26,32,30,34)(28,35,37,38)(33,39,36,40);(11,18)(14,15)(19,24)(22,23)(26,30)(32,34)(33,36)(39,40);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,29,31,27)(26,34,30,32)(28,35,37,38)(33,39,36,40);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,30,31,26)(27,34,29,32)(28,36,37,33)(35,40,38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7979502);