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 | #35365 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22)(26,31)(29,39)(32,40)(33,36);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,18)(13,20,19,22)(14,21,24,23)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(10,17)(13,19)(16,18)(20,22)(26,31)(28,38)(32,40)(34,35);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,18,17,16)(13,22,19,20)(14,23,24,21)(25,27,30,37)(26,32,31,40)(28,35,38,34)(29,36,39,33);(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,21,18,23)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,22)(18,20)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|b^2=hi,d^2=g,e^2=h,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, 7549751);