Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420407 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,7)(9,10)(11,19)(12,23)(13,17)(14,27)(15,21)(16,25)(18,26)(20,34)(22,30)(24,38)(28,33)(29,40)(31,36)(32,39)(35,37);(1,2,4,5)(3,7,8,6)(9,11,13,21)(10,15,17,19)(12,20,24,35)(14,28,26,22)(16,31,32,29)(18,33,27,30)(23,37,38,34)(25,40,39,36);(9,12)(10,16,17,32)(11,20)(13,24)(14,25)(15,29,19,31)(18,23,27,38)(21,35)(22,36)(26,39)(28,40)(30,37,33,34);(1,3,4,8)(2,6,5,7)(11,21)(14,26)(15,19)(18,27)(20,35)(23,38)(25,39)(29,31)Polycyclic presentation:
2^9|b^2=e,c^2=g,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 350647);