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 | #3147 | C22 |
| 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 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,13,19,15,20,14,30)(11,16,22,32,24,33,23,18)(12,25,27,31,29,40,28,35)(17,34,36,26,37,38,39,21);(1,2)(3,7)(4,8)(5,6)(9,11,13,22,15,24,14,23)(10,16,19,32,20,33,30,18)(12,26,27,38,29,21,28,34)(17,35,36,25,37,31,39,40);(1,3,5,4)(2,7,6,8)(9,12,23,38,14,28,24,26,15,29,22,34,13,27,11,21)(10,17,16,31,19,36,32,40,20,37,33,35,30,39,18,25);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=efh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405806);