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 | C22×#102 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
855 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)(9,10,13,18,14,19,28,36,15,20,29,37,27,35,12,17)(11,21,23,33,24,34,22,38,25,39,26,16,30,31,40,32);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8)(9,11)(10,16)(12,26)(13,22)(14,30)(15,25)(17,34)(18,21)(19,38)(20,33)(23,28)(24,27)(29,40)(31,36)(32,35)(37,39);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,f^c=fh,g^c=gh,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403669);