Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C16 |
| Frattini, cosocle | #1954 | C22 |
| socle | C22 | #1316 |
| prime radical | C2×H(5,𝔽2) | C8 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1954 ◁ G | |
| LECS | G ▷ #1954 ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18)(19,26,27,22)(20,24,30,23)(21,25)(28,32,29,31);(17,19)(18,22)(20,29)(21,28)(23,26)(24,32)(25,31)(27,30)Polycyclic presentation:
2^9|a^2=d,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=dei,e^b=ei,g^b=gi,d^c=dgi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 873);