Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | C23 | C2×#135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,21,29,20)(11,16,13,28,23,18,30,32)(12,24,27,37,25,31,40,38)(17,33,35,39,34,22,36,26);(2,5)(3,7)(9,11,14,13,15,23,29,30)(10,16,19,28,21,18,20,32)(12,22,27,26,25,33,40,39)(17,31,35,38,34,24,36,37);(9,12,23,33,14,27,30,39,15,25,11,22,29,40,13,26)(10,17,28,38,20,36,16,31,21,34,32,37,19,35,18,24);(1,3,4,7)(2,6,5,8)(12,25)(17,34)(22,33)(24,31)(26,39)(27,40)(35,36)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,d^c=di86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405733);