Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 conjugacy classes of subgroups; only the 236 normal subgroups are shown.
This group has 942 conjugacy classes of subgroups — too many to draw. Only its 236 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:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(25,26)(27,35)(28,32)(29,34)(30,33)(31,37)(36,40)(38,39);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,18,20,22)(19,23,24,21)(25,27,29,37)(26,31,34,35)(28,38,30,36)(32,40,33,39);(3,7)(6,8)(10,14)(13,16)(17,19,20,24)(18,21,22,23)(25,28,29,30)(26,32,34,33)(27,36,37,38)(31,39,35,40);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(26,33)(27,37)(28,30)(31,40)(32,34)(35,39)Polycyclic presentation:
2^9|a^2=ei,b^2=g,c^2=g,d^2=ei,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, 386801);