Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484587 |
| 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 | |
971 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 971 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(2,6)(3,4)(7,8)(9,10)(11,17)(12,23)(13,18)(14,19)(15,20)(16,22)(21,32)(24,34)(25,31)(26,35)(27,36)(28,29)(30,37)(33,38)(39,40);(1,2)(3,6)(4,8)(5,7)(9,11,15,12)(10,16,20,31)(13,24,27,21)(14,22,28,25)(17,29,23,19)(18,33,36,30)(26,38,39,37)(32,35,34,40);(1,3,5,4)(2,6,7,8)(9,12,15,11)(10,17)(13,21,27,24)(14,25,28,22)(16,29)(18,34)(19,31)(20,23)(26,37,39,38)(30,35)(32,36)(33,40);(9,13,15,27)(10,18,20,36)(11,21,12,24)(14,26,28,39)(16,30,31,33)(17,32,23,34)(19,35,29,40)(22,37,25,38)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,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,d^c=di,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, 391985);