Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×C22⋊C16 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,29,24,31,46,58,25,28,42,27,47,56,70,57,23,30,40,55,39,54,71,77,21,44,20,38,41,68,43,69)(19,32,35,59,37,60,67,78,34,26,48,50,65,52,72,76,36,53,66,49,62,74,80,75,45,51,33,61,63,73,64,79);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,36)(18,26,30,51)(20,33,42,48)(21,45,25,34)(22,35,40,66)(24,37,39,62)(27,49,38,59)(28,53,44,32)(29,50,55,61)(31,52,54,73)(41,63,47,65)(43,64,70,72)(46,67,71,80)(56,74,68,60)(57,75,69,78)(58,76,77,79);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,39,70,25,22,41,71,23,42,24,43,21,40,47,46)(18,27,54,69,28,55,68,58,30,38,31,57,44,29,56,77)(19,33,62,72,34,35,63,80,36,48,37,64,45,66,65,67)(26,49,73,78,53,61,60,76,51,59,52,75,32,50,74,79)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=fg,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=di,e^b=efi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56933);