Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
470 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,9,11,12)(8,10)(13,14,16,21,18,22,26,25,17,20,23,28,27,15,19,24);(5,7,8,11)(6,9)(10,12)(13,15,23,24,18,14,19,21,17,22,16,25,27,20,26,28);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dg,f^b=fg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47817);