Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C4×C8 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C22 | #1326 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(5,6,9,15,11,17,34,48,12,18,35,49,10,16,33,47)(7,19,21,50,23,52,56,67,24,42,57,64,22,51,31,61)(8,25,27,58,29,14,41,43,30,44,32,62,28,45,60,68)(13,36,37,20,39,53,46,55,40,63,66,26,38,59,65,54);(1,2,3,4)(5,7,10,22,12,24,11,23)(6,13,16,38,18,40,17,39)(8,26,28,55,30,20,29,54)(9,31,33,57,35,56,34,21)(14,42,25,52,45,19,44,51)(15,46,47,37,49,65,48,66)(27,36,60,59,32,63,41,53)(43,67,58,50,68,61,62,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1293);