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 | |
127 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 58 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)(17,18,21,27,24,30,47,31)(19,32,34,67,37,57,38,76)(20,39,42,65,44,28,63,62)(22,48,45,60,43,26,56,59)(23,50,46,29,49,58,41,64)(25,51,52,33,55,69,61,40)(35,66,70,54,72,78,73,79)(36,74,68,77,71,80,75,53);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,35,49,71,63,51,47,34,56,70,46,75,44,33,24,37,43,72,23,36,42,69,21,38,45,73,41,68,20,40)(18,25,28,53,64,79,59,67,27,61,62,80,50,78,48,32,30,55,39,77,29,54,60,76,31,52,65,74,58,66,26,57)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=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, 1405);