Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C8 |
| Frattini, cosocle | C4⋊#95 | C22 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C4⋊#95 ◁ G | |
| LECS | G ▷ C4⋊#95 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
469 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,13,12,14,15,16)(17,18)(19,26,27,22)(20,24,30,23)(21,25)(28,32,29,31);(2,5)(3,8)(6,7)(17,19)(18,22)(20,29)(21,28)(23,26)(24,32)(25,31)(27,30)Polycyclic presentation:
2^9|a^2=d,c^2=f,d^2=g,e^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.
| maximal subgroup of | #72267798, #72267844 |
| maximal normal subgroup of | #72267798, #72267844 |
| maximal quotient of | #72267802, #72267811, #72267812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1101);