Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,30,33)(27,34,38,35)(28,39,32,29)(31,36,40,37);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,29,37,30,38,39,36)(26,31,32,34,33,40,28,35);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(12,19)(20,24)(25,28,29,26,30,32,39,33)(27,35,37,31,38,34,36,40);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,29,30,39)(26,28,33,32)(27,36,38,37)(31,34,40,35);(11,17)(14,21)(18,23)(20,24)(25,29,30,39)(26,28,33,32)(27,37,38,36)(31,35,40,34)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433725);