Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10,11,16,15,19,14,29)(12,22,24,36,25,37,20,35)(13,26,21,34,28,40,27,18)(17,31,32,23,33,38,30,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,19,29)(12,20,25,24)(13,21,28,27)(17,30,33,32)(18,26,34,40)(22,35,37,36)(23,31,39,38);(2,5)(6,8)(9,12,13,23,14,24,27,38,15,25,28,39,11,20,21,31)(10,17,18,22,16,30,26,35,19,33,34,37,29,32,40,36);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310095, #72310107, #72310108, … |
| maximal normal subgroup of | #72310095, #72310107, #72310108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403926);