Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
762 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 762 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)(11,15)(13,16)(17,18,21,20)(19,23,24,22)(25,26)(27,30)(28,32)(29,31)(33,36)(34,37)(35,38)(39,40);(1,3,6,2)(4,8,5,7)(9,10,12,14)(11,16,15,13)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,37)(33,39)(36,40);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,23,24,22)(27,34)(30,37)(35,40)(38,39);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(18,20)(19,24)(26,31)(29,36)(30,38)(32,33)(35,40)(37,39)Polycyclic presentation:
2^9|a^2=efgi,b^2=f,c^2=ef,d^2=fgi,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, 386926);