Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94206 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
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 | #358 |
| LCS | G ▷ C2×C16 ▷ 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 | |
131 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 131 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)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,27,24,29,25,33)(19,20,30,35,32,37,39,38)(22,40,36,26,41,47,42,31)(23,43,34,48,44,46,45,28);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,28,23,31,20,33,21,39,36,46,34,47,35,29,24,32,41,48,44,26,37,27,25,30,42,43,45,40,38,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgh,d^2=i,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=ci,d^b=dg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1404);