Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
511 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,15)(11,18)(12,14)(16,23)(19,24)(20,21)(25,26,28,32,29,31,35,38)(27,30,34,37,33,36,39,40);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,19,21,16)(12,22,14,17)(18,23,20,24)(27,33)(30,36)(34,39)(37,40);(3,8)(5,7)(9,11,12,20,13,21,14,18)(10,16,22,23,15,19,17,24)(25,27)(26,30)(28,34)(29,33)(31,36)(32,37)(35,39)(38,40);(2,6)(5,7)(9,12,13,14)(10,17,15,22)(11,20,21,18)(16,24,19,23)(26,31)(27,33)(32,38)(34,39)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 124599);