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 | C24 | C22×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
854 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 854 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:
(1,2,4,7)(3,5,8,6)(9,10)(11,20)(12,18)(13,17)(14,19)(15,21)(16,25)(22,35)(23,28)(24,33)(26,40)(27,34)(29,36)(30,37)(31,39)(32,38);(2,5)(3,8)(6,7)(9,11,14,23)(10,15,19,30)(12,21,25,37)(13,26,27,22)(16,28,18,20)(17,32,34,29)(24,38,39,36)(31,35,33,40);(9,12,14,25)(10,16)(11,21,23,37)(13,24,27,39)(15,28)(17,31)(18,19)(20,30)(22,36,26,38)(29,40)(32,35)(33,34);(9,13,14,27)(10,17,19,34)(11,22,23,26)(12,24,25,39)(15,29,30,32)(16,31,18,33)(20,35,28,40)(21,36,37,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,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, 391863);