Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94343 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1380 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #367 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18,21,26,24,30,38,29)(19,31,32,48,27,20,35,37)(22,42,40,34,43,47,39,25)(23,44,41,28,45,46,36,33);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,22,33,23,34,20,29,38,32,39,44,36,47,48,18,24,27,43,28,45,25,31,26,21,35,40,46,41,42,37,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=gi,e^2=f,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,h^a=hi,c^b=cg,d^b=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1601);