Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94202 |
| 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 | |
129 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 129 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) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 80 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,28,24,31,25,50)(19,32,34,64,37,65,43,66)(20,29,39,59,41,62,42,76)(22,45,44,61,40,27,47,58)(23,48,38,60,46,57,49,30)(26,51,52,33,55,68,56,70)(35,73,71,54,69,63,77,79)(36,75,72,53,74,78,67,80);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,35,46,74,39,51,21,43,44,77,49,72,41,70,24,37,40,69,23,36,42,68,25,34,47,71,38,67,20,33)(18,26,29,53,60,79,58,66,50,52,76,78,48,73,27,32,31,55,62,80,30,54,61,64,28,56,59,75,57,63,45,65)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgh,d^2=i,e^2=fi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,h^a=hi,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1401);