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 |
| prime radical | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,19,20,14)(12,21,24,17)(13,23,22,18)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,5,4)(3,6,7,8)(9,11,15,20)(10,14,16,19)(12,22,24,13)(17,18,21,23)(28,36)(31,38)(34,40)(37,39);(2,4)(6,8)(10,16)(14,19)(17,21)(18,23)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(3,7)(6,8)(9,12,15,24)(10,17,16,21)(11,13,20,22)(14,23,19,18)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,12)(14,21)(15,22)(16,23)(17,19)(20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,4,5,2)(3,8,7,6)(9,14,15,19)(10,11,16,20)(12,23,24,18)(13,21,22,17)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231996);