Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C22 | #1326 |
| prime radical | C24 | #101 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ #1729 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(2,5,6,7)(9,10,13,19,15,22,35,49,16,21,36,50,14,20,38,52)(11,23,25,53,28,54,34,67,27,55,59,72,26,60,62,70)(12,29,31,63,32,46,65,48,33,51,66,71,37,18,44,45)(17,39,40,24,42,56,47,57,43,64,68,30,41,58,69,61);(1,2)(3,7)(4,6)(5,8)(9,11,14,26,16,27,15,28)(10,17,20,41,21,43,22,42)(12,30,32,61,33,24,37,57)(13,34,35,59,36,62,38,25)(18,23,29,54,46,55,51,60)(19,47,49,68,50,69,52,40)(31,64,44,56,66,39,65,58)(45,70,71,72,48,67,63,53)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=defhi,f^b=fi,h^b=hi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1148);