Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
480 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)(3,5)(4,6)(7,8)(9,10,15,21,16,22,13,20,14,18,30,37,32,12,19,31)(11,23,27,17,28,36,33,29,26,24,38,35,34,40,25,39);(1,3,4,7)(2,5,6,8)(9,11,14,26)(10,17,18,35)(12,29,22,39)(13,27,19,38)(15,33,30,25)(16,34,32,28)(20,36,31,40)(21,24,37,23);(2,6)(5,8)(10,18)(12,22)(17,35)(20,31)(21,37)(23,24)(29,39)(36,40);(9,12,30,20,16,10,19,37,14,22,15,31,32,18,13,21)(11,24,27,35,28,40,33,39,26,23,38,17,34,36,25,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404963);