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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(10,15)(13,17)(16,22)(20,23)(25,26)(27,31)(28,37)(29,39)(30,34)(32,38)(33,35)(36,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,30,36)(26,31,34,40)(28,35,38,29)(32,39,37,33);(10,15)(12,18)(16,22)(19,24)(25,26,30,34)(27,31,36,40)(28,32,38,37)(29,33,35,39);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,16,15,22)(12,19,18,24)(13,20,17,23)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,40)(34,37);(1,4,5,3)(2,7,8,6)(9,12,14,18)(10,17,15,13)(11,19,21,24)(16,23,22,20)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(9,13,14,17)(10,18,15,12)(11,20,21,23)(16,24,22,19)(25,29,30,35)(26,33,34,39)(27,28,36,38)(31,32,40,37)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550620);