Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,10,11,12)(13,14,17,19,18,15,16,20);(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,15,18,14)(16,19,17,20);(5,6)(7,10)(8,12)(9,11)(13,14,18,15)(16,19,17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,f^b=fhi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13061);