Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6,10,11)(7,9,12,8)(13,14,17,24,19,26,33,40,20,22,18,25,16,23,32,39)(15,27,30,42,29,37,43,41,28,38,34,44,31,21,35,36);(2,4)(5,7,9,11,10,12,8,6)(13,15,18,30,19,29,32,43,20,28,17,34,16,31,33,35)(14,21,25,36,26,27,39,42,22,37,24,41,23,38,40,44);(5,8,10,9)(6,7,11,12)(14,22)(15,28)(23,26)(24,25)(29,31)(30,34)(35,43)(39,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fg,c^b=cgi,d^b=di,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34025);