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 | #1316 |
| prime radical | C24 | #100 |
| 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 | |
272 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 272 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)(9,10,13,19,16,20,34,53,15,21,35,52,14,36,39,51)(11,22,24,54,25,55,60,71,26,46,59,68,37,67,32,64)(12,27,29,61,30,18,45,47,31,49,33,65,38,48,63,72)(17,40,41,23,44,56,50,58,43,66,69,28,42,62,70,57);(1,2,4,6)(3,7,8,5)(9,11,15,26)(10,17,21,43)(12,28,31,23)(13,32,35,60)(14,37,16,25)(18,46,48,22)(19,50,52,70)(20,44,36,42)(24,39,59,34)(27,55,49,67)(29,40,33,66)(30,57,38,58)(41,53,69,51)(45,56,63,62)(47,71,72,64)(54,65,68,61)Polycyclic presentation:
2^9|a^2=d,b^2=f,c^2=f,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defi,e^b=ei,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, 1175);