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 | |
4581 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)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3,5,8)(2,4,6,7)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(27,38)(28,39)(32,37)(33,40);(1,2)(3,4)(5,6)(7,8)(25,27,31,38)(26,32,36,37)(28,35,39,29)(30,33,34,40);(1,2)(3,7)(4,8)(5,6)(9,12)(10,16)(11,13)(14,20)(15,17)(18,22)(19,24)(21,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,34)(36,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,29,31,35)(26,34,36,30)(27,28,38,39)(32,40,37,33);(1,4,5,7)(2,3,6,8)(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,30,31,34)(26,35,36,29)(27,33,38,40)(28,32,39,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^2=i,f^2=h,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, 8356552);