Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
664 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 664 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,6)(4,7)(5,8)(9,10,14,20,16,21,15,31)(11,17,23,19,13,30,24,36)(12,25,27,39,29,40,28,32)(18,33,34,22,35,37,38,26);(1,3,6,2)(4,7,5,8)(9,11,14,23,16,13,15,24)(10,17,20,19,21,30,31,36)(12,22,27,37,29,26,28,33)(18,32,34,25,35,39,38,40);(1,4,6,5)(2,7,3,8)(9,12,23,37,15,28,11,22,16,29,24,33,14,27,13,26)(10,18,30,39,20,34,36,40,21,35,17,32,31,38,19,25);(1,5,6,4)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404334);