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 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,16,15)(11,20,22,17)(12,18,13,23)(14,24,21,19)(25,26)(27,31)(28,32)(29,40)(30,34)(33,36)(35,37)(38,39);(1,3,5,8)(2,4,6,7)(9,11)(10,17)(12,14)(13,21)(15,20)(16,22)(18,19)(23,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,2)(3,4)(5,6)(7,8)(9,12,16,13)(10,18,15,23)(11,14,22,21)(17,19,20,24)(25,27,30,37)(26,31,34,35)(28,36,39,29)(32,33,38,40);(1,2)(3,7)(4,8)(5,6)(9,13,16,12)(10,18,15,23)(11,21,22,14)(17,19,20,24)(25,28)(26,32)(27,36)(29,37)(30,39)(31,33)(34,38)(35,40);(9,14,16,21)(10,19,15,24)(11,12,22,13)(17,18,20,23)(25,29,30,36)(26,33,34,40)(27,28,37,39)(31,38,35,32);(1,4,5,7)(2,3,6,8)(9,15,16,10)(11,20,22,17)(12,23,13,18)(14,24,21,19)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=ghi,f^2=g,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, 8356498);