Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #3459 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,13,19,14,20,29,37,15,17,28,33,12,18,26,35)(11,21,23,38,22,31,39,36,24,34,27,40,25,16,30,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,31)(18,34)(19,36)(20,21)(23,28)(26,39)(29,30)(32,33)(35,38)(37,40);(3,7)(5,8)(10,17)(13,28)(16,31)(18,20)(21,34)(23,27)(26,29)(30,39)Polycyclic presentation:
2^9|a^2=f,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dgi,e^b=eg,f^b=fi,d^c=dg,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36857);