Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 80 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,21,27,44,59,69,57,24,30,20,37,23,29,45,61,22,28,42,60,39,55,41,65,47,62,40,26,46,56,77,80)(19,31,33,66,72,58,76,25,36,50,32,68,35,54,73,53,34,67,71,52,48,78,38,51,49,79,70,63,74,75,43,64);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,34)(18,25,28,51)(20,38,40,76)(21,43,42,73)(23,48,46,72)(24,49,47,36)(26,54,37,75)(27,58,60,78)(29,63,56,68)(30,64,62,53)(31,65,67,57)(32,69,70,41)(33,45,71,77)(35,44,74,39)(50,61,79,80)(52,59,66,55);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fi,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58164);