Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1560 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 1560 conjugacy classes of subgroups — too many to draw. Only its 274 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:
(1,2)(3,4)(13,14)(15,19,20,17)(16,18)(21,22,23,25,24,26,27,28);(5,6,8,10)(7,11,12,9)(13,15,16,20)(14,17,18,19);(2,4)(5,7,8,12)(6,9,10,11)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=f,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29175);