Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94343 |
| 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 | #366 |
| 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 | |
117 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 117 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:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18)(19,31,33,59)(20,27)(21,41)(22,44)(23,47)(24,30)(25,32,48,62)(26,38)(28,46)(29,45)(34,67,63,58)(35,69,65,80)(36,71,66,60)(37,54)(39,74)(40,57)(42,52)(43,53)(49,61,76,70)(50,78,77,68)(51,79,75,64)(55,73)(56,72);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,21,34,42,65,72,79,45,66,73,78,46,70,40,62,24,33,38,63,22,35,39,64,23,36,43,68,37,61,20,32)(18,25,27,49,54,77,53,60,47,75,74,80,44,67,26,31,30,48,57,76,28,50,55,71,29,51,56,69,52,58,41,59)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,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=cg,d^b=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1600);