Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484583 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
840 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,7)(6,8)(9,10)(11,21)(12,19)(13,18)(14,17)(15,20)(16,22)(23,37)(24,36)(25,34)(26,40)(27,32)(28,35)(29,31)(30,38)(33,39);(1,2,4,6)(3,8,7,5)(9,11,15,24)(10,16,20,31)(12,22,14,29)(13,26,28,23)(17,21,19,36)(18,33,35,30)(25,39,27,38)(32,40,34,37);(9,12)(10,17,20,19)(11,22)(13,25)(14,15)(16,21,31,36)(18,32,35,34)(23,38)(24,29)(26,39)(27,28)(30,37,33,40);(9,13,15,28)(10,18,20,35)(11,23,24,26)(12,25,14,27)(16,30,31,33)(17,32,19,34)(21,37,36,40)(22,38,29,39)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391897);