Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C22×C4 | C2×C16 |
| Frattini, cosocle | C2×#336 | C22 |
| socle | C23 | #324 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C2×#336 ▷ C23×C4 ▷ C22 ▷ C1 | |
306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(2,5,6,7)(9,10)(11,14,16,13)(12,15)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2)(3,7)(4,6)(5,8)(10,13,15,14)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fh,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defh,d^c=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68);