Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
376 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 3 generators on 32 points:
(1,2)(3,11)(4,8)(5,9,14,15)(6,10)(7,12,16,13)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,3,6,4)(2,7)(5,12,14,13)(8,15)(9,11)(10,16);(1,4,6,3)(2,8,10,11)(5,13,14,12)(7,15,16,9)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=h,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,e^b=efh,e^d=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29492);