Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | #2285 | C22 |
| socle | C22 | #1390 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #2285 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
298 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,18,15,19,14,27)(11,20,22,37,23,38,28,33)(12,17,25,34,26,36,29,35)(16,30,31,39,32,24,40,21);(2,5,7,6)(9,11)(10,16,18,31,19,32,27,40)(12,24)(13,22)(14,28)(15,23)(17,33,35,38,36,37,34,20)(21,25)(26,30)(29,39)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,h^a=hi,c^b=cg,g^b=gh,e^c=ehi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1522);