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 | |
4596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,31)(27,28,33,32)(29,34,38,39)(35,36,40,37);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(26,31)(27,33)(34,39)(35,40);(1,3)(2,5)(4,8)(6,7)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,3)(2,6)(4,8)(5,7)(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,f^2=g,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, 8003089);