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 | C22×#135 | C22 |
| 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 | |
660 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)(9,10,14,18,15,12,26,19)(11,20,23,36,25,39,24,21)(13,29,32,17,33,31,37,28)(16,34,35,40,27,30,38,22);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,27)(13,30)(14,23)(15,25)(17,36)(18,35)(19,38)(20,29)(21,28)(22,32)(24,26)(31,39)(33,34)(37,40);(2,6)(5,8);(9,12)(10,15)(11,21)(13,31)(14,19)(16,34)(17,37)(18,26)(20,23)(22,38)(24,39)(25,36)(27,30)(28,32)(29,33)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404929);