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 | |
4630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10,16,15)(11,19,14,17)(12,18,13,22)(20,24,21,23)(25,26,31,35)(27,32,38,36)(28,30,34,40)(29,39,37,33);(1,2)(3,7)(4,8)(5,6)(9,11,16,14)(10,17,15,19)(12,20,13,21)(18,23,22,24)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,37,39);(1,3)(2,7)(4,6)(5,8)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,27,31,38)(26,32,35,36)(28,37,34,29)(30,33,40,39);(1,3)(2,4)(5,8)(6,7)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,28)(26,30)(27,37)(29,38)(31,34)(32,33)(35,40)(36,39);(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23)(25,29,31,37)(26,33,35,39)(27,28,38,34)(30,32,40,36);(1,4)(2,8)(3,6)(5,7)(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,30)(26,34)(27,33)(28,35)(29,32)(31,40)(36,37)(38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 8365508);