Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
864 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 212 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) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,6,3)(4,8,5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,22)(21,23)(25,26)(27,31)(28,36)(29,34)(30,33)(32,37)(35,40)(38,39);(1,3,6,2)(4,8,5,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,21)(23,24)(25,27,29,30)(26,31,34,33)(28,35,37,38)(32,39,36,40);(1,4,6,5)(2,7,3,8)(11,15)(13,16)(17,20,21,19)(18,22,23,24)(25,28,29,37)(26,32,34,36)(27,35,30,38)(31,39,33,40);(1,5,6,4)(2,7,3,8)(19,20)(22,24)(26,33)(27,30)(31,34)(32,40)(35,38)(36,39)Polycyclic presentation:
2^9|a^2=efgi,b^2=f,c^2=ef,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386725);