Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94533 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7344 |
| commutator | C4⋊Q8 | C2×C8 |
| Frattini, cosocle | #1963 | C22 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,7,12,28,43,6,13,30,44,31,11,29,42)(3,14,16,46,18,47,51,64,17,37,26,59,19,52,54,60)(4,20,22,55,24,39,57,63,23,45,58,41,25,9,27,38)(8,32,33,15,35,48,61,49,36,50,62,21,34,56,40,53);(1,3,6,17)(2,8,13,36)(4,21,23,15)(5,26,30,16)(7,19,31,18)(9,37,39,14)(10,40,44,61)(11,35,12,34)(20,47,45,52)(22,48,58,56)(24,49,25,53)(27,50,57,32)(28,51,29,54)(33,43,62,42)(38,60,63,64)(41,46,55,59)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=i,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=dehi,e^b=eh,f^b=fi,g^b=gh,d^c=dghi,e^c=eh,g^c=gi,e^d=ehi,g^d=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1733);