Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395181 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
964 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 226 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:
(9,10)(11,20)(12,24,26,31)(13,18)(14,19)(15,22)(16,25,32,38)(17,27)(21,28,35,39)(23,29)(30,36)(33,34,37,40);(1,2,4,6)(3,7,8,5)(9,11,13,22,14,23,27,36)(10,15,17,20,19,30,18,29)(12,21,25,34,26,35,38,40)(16,28,31,33,32,39,24,37);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,17)(14,19)(18,27)(20,22)(21,28)(23,30)(24,38)(25,31)(26,32)(29,36)(33,34)(35,39)(37,40);(9,12,13,25,14,26,27,38)(10,16,17,31,19,32,18,24)(11,21,22,34,23,35,36,40)(15,28,20,33,30,39,29,37)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310867);