Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C4×C8 |
| Frattini, cosocle | C4×#124 | C22 |
| socle | C22 | #1390 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 260 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,4,7,6,8,5,3)(9,10,11,15)(12,21,22,18)(13,17,20,24)(14,23,16,19);(1,3,5,8,6,7,4,2)(9,10)(11,18)(12,21)(13,17)(14,20)(15,19)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=g,f^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1328);