Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150198 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #382 | C23 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19,21,36,22,35,26,37)(12,23,24,34)(13,18)(14,16)(15,20,30,38,29,40,33,39)(17,25,31,27)(28,32);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,19)(18,33)(21,28)(23,36)(24,40)(25,38)(27,39)(30,32)(31,35)(34,37);(10,16)(15,29)(17,31)(18,32)(19,35)(23,34)(30,33)(36,37)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57430);