Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,15,21,28,20)(11,22,23,36,26,37,39,31)(13,19,14,18,27,34,29,35)(16,25,30,38,33,40,32,24);(1,3)(2,5)(4,8)(6,7)(9,11,14,25,12,23,27,38,15,26,29,40,28,39,13,24)(10,16,19,31,20,32,35,37,21,33,34,36,17,30,18,22);(9,12,15,28)(10,17,21,20)(11,23,26,39)(13,14,27,29)(16,30,33,32)(18,34,35,19)(22,36,37,31)(24,25,38,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=ghi,e^2=i,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,e^b=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34521);