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 | |
4314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,21)(14,16)(15,18)(17,24)(19,23)(20,22);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,22)(15,16)(17,20)(19,24)(21,23)(26,30)(31,37)(32,38)(39,40);(9,12,15,14)(10,16,18,11)(13,20,24,23)(17,22,21,19)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,19)(12,20)(14,23)(15,24)(16,22)(18,21);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,23)(22,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(1,3)(2,7)(4,6)(5,8)(9,14,15,12)(10,16,18,11)(13,23,24,20)(17,22,21,19)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003516);