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 | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,12,18,15,19,27,34,14,17,26,32,13,28,25,38)(11,16,21,29,23,30,36,39,20,31,24,33,22,37,35,40);(1,2,4,8,5,7,3,6)(9,11,13,22,14,20,15,23)(10,16,19,30,17,31,28,37)(12,24,25,36,26,21,27,35)(18,33,34,40,32,29,38,39);(2,7)(6,8)(10,17)(11,20)(18,32)(19,28)(21,24)(22,23)(34,38)(35,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fg,c^b=cg,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, 33880);