Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233469 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
698 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 698 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,12,15)(11,17,19,14)(13,16,21,24)(18,23,20,22)(25,26,29,35,31,34,30,36)(27,37,39,28,38,40,33,32);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,21)(23,24)(25,27,31,38)(26,32,34,28)(29,33,30,39)(35,40,36,37);(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,28,31,32)(26,33,34,39)(27,35,38,36)(29,37,30,40);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,24)(22,23)(26,34)(27,38)(33,39)(35,36)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=h,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151557);