Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| 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 | |
193 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 88 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)(17,18,22,29,24,31,46,60,25,28,42,56,47,58,70,77,23,30,40,59,39,54,71,78,21,44,20,27,41,55,43,57)(19,32,35,53,37,61,66,76,34,26,33,49,64,51,65,75,36,52,63,48,62,73,79,74,45,50,38,67,68,72,69,80);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,36)(18,26,30,50)(20,38,42,33)(21,45,25,34)(22,35,40,63)(24,37,39,62)(27,53,56,48)(28,52,44,32)(29,49,59,67)(31,51,54,72)(41,68,47,64)(43,69,70,65)(46,66,71,79)(55,61,58,73)(57,76,77,74)(60,75,78,80);(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)(18,27,54,77,28,29,55,78,30,56,31,57,44,59,58,60)(19,33,62,69,34,63,68,66,36,38,37,65,45,35,64,79)(26,48,72,76,52,67,61,75,50,53,51,74,32,49,73,80)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,c^b=ci,d^b=di,e^b=efi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56934);