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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(26,31)(29,39)(32,38)(33,40);(1,2,5,6)(3,7,4,8)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(10,14)(15,17)(16,20)(22,23)(26,31)(28,37)(32,38)(34,35);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27,30,36)(26,32,31,38)(28,35,37,34)(29,33,39,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,40)(32,39)(34,36);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,38,37,32)Polycyclic presentation:
2^9|b^2=gi,d^2=gh,f^2=g,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, 7549711);