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 | |
4243 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)(3,8)(4,6)(5,7)(9,10,15,17)(11,19,14,16)(12,18,22,13)(20,24,23,21)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,33,36,40);(1,3,5,6)(2,4,7,8)(9,11,15,14)(10,16,17,19)(12,20,22,23)(13,24,18,21)(25,27,30,37)(26,31,34,35)(28,29,39,36)(32,33,38,40);(10,17)(12,22)(16,19)(20,23)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,40,35,33);(9,12)(10,18)(11,20)(13,17)(14,23)(15,22)(16,21)(19,24)(25,28)(26,32)(27,29)(30,39)(31,33)(34,38)(35,40)(36,37);(1,2)(3,4)(5,7)(6,8)(9,13,15,18)(10,12,17,22)(11,21,14,24)(16,23,19,20)(25,29)(26,33)(27,28)(30,36)(31,32)(34,40)(35,38)(37,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,19,17,16)(12,23,22,20)(13,21,18,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9627578);