Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149899 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | #3486 | C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 449 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,23,11,14,10,28)(3,13,15,30)(4,18)(8,22,26,24,12,25,27,29)(16,31,17,19)(20,32);(1,2)(3,14)(4,9)(5,8)(6,11)(7,17)(10,20)(12,21)(13,24)(15,22)(16,25)(18,26)(19,29)(23,31)(27,32)(28,30);(2,8)(7,22)(9,26)(10,27)(11,12)(14,25)(23,24)(28,29)Polycyclic presentation:
2^9|a^2=e,d^2=i,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cgh,d^b=di,e^b=ef,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57095);