Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #384 | C23 |
| socle | C22 | C2×AΣL(1,𝔽16) |
| 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 ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 398 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:
(2,5,6,7)(9,10)(11,20,23,29,24,33,27,15)(12,16)(13,17)(14,19)(18,21)(22,34,36,39,37,40,38,28)(25,30)(26,32)(31,35);(1,2)(3,7)(4,6)(5,8)(9,11,21,23,14,24,13,27)(10,15,17,20,19,29,18,33)(12,22,35,36,26,37,25,38)(16,28,30,34,32,39,31,40);(9,12,14,26)(10,16,19,32)(11,22,24,37)(13,25,21,35)(15,28,29,39)(17,30,18,31)(20,34,33,40)(23,36,27,38)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=i,d^2=gh,e^2=hi,f^2=i,b^a=bd,d^a=df,f^a=fh,d^b=dg,e^b=efghi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57511);