Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | 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×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 90 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)(17,18,22,29,24,31,45,59,25,28,41,55,46,57,68,76,23,30,39,58,38,53,69,77,21,43,20,27,40,54,42,56)(19,32,35,48,37,61,67,73,34,26,47,49,65,51,70,74,36,52,66,72,62,75,79,80,44,50,33,60,63,71,64,78);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,36)(18,26,30,50)(20,33,41,47)(21,44,25,34)(22,35,39,66)(24,37,38,62)(27,48,55,72)(28,52,43,32)(29,49,58,60)(31,51,53,71)(40,63,46,65)(42,64,68,70)(45,67,69,79)(54,61,57,75)(56,73,76,80)(59,74,77,78);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,38,68,25,22,40,69,23,41,24,42,21,39,46,45)(18,27,53,76,28,29,54,77,30,55,31,56,43,58,57,59)(19,33,62,70,34,35,63,79,36,47,37,64,44,66,65,67)(26,48,71,80,52,49,61,78,50,72,51,73,32,60,75,74)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,d^a=df,d^b=di,e^b=efi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56932);