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 |
| prime radical | #47800 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4287 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)(9,10,15,17)(11,18,13,16)(12,14,19,24)(20,23,22,21)(26,31)(28,32)(34,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,18)(12,20,19,22)(14,21,24,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(1,3,5,4)(2,7,6,8)(10,17)(11,13)(14,24)(20,22)(26,31)(28,32)(34,39)(36,40);(9,12)(10,14)(11,20)(13,22)(15,19)(16,21)(17,24)(18,23)(25,27)(26,32)(28,31)(29,37)(30,33)(34,36)(35,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,18,17,16)(12,22,19,20)(14,23,24,21)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35);(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,22)(18,20)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,e^2=g,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, 7982464);