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 | |
4444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10,15,18)(11,19,22,16)(12,23,21,14)(13,24,20,17)(26,31)(28,34)(32,38)(36,40);(1,3,5,7)(2,4,8,6)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,19)(23,24)(27,35)(32,38)(36,40)(37,39);(1,3)(2,6)(4,8)(5,7)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(9,12,15,21)(10,14,18,23)(11,20,22,13)(16,17,19,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(9,13,15,20)(10,17,18,24)(11,21,22,12)(14,19,23,16)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(1,4)(2,7)(3,8)(5,6)(9,14)(10,12)(11,17)(13,19)(15,23)(16,20)(18,21)(22,24)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8370762);