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 | |
4493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(29,37)(34,38)(35,40)(36,39);(1,2,5,3)(4,7,8,6)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,5,2)(4,7,8,6)(10,12)(15,17)(16,20)(22,23)(26,31)(28,32)(34,38)(36,39);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,30,33)(26,32,31,28)(29,35,37,40)(34,39,38,36);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002989);