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 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,17)(13,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,31)(28,30)(29,32)(33,35)(34,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,14,16,17)(13,19,22,20)(18,21,23,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,40,38,39);(10,16)(14,17)(18,23)(21,24)(26,28)(29,37)(31,33)(34,38);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,17,16,14)(13,19,22,20)(18,21,23,24)(25,27)(26,31)(28,33)(29,38)(30,35)(32,40)(34,37)(36,39);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(9,14,15,17)(10,11,16,12)(13,21,22,24)(18,19,23,20)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|b^2=h,d^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7998944);