Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2188 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 2188 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,21)(16,24,25,19)(17,20)(18,22,26,28)(23,27);(13,15,17,23)(14,18,20,26)(16,22,25,28)(19,21,24,27);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fg,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284451, #72285100, #72285142, … |
| maximal normal subgroup of | #72284451, #72285100, #72285142, … |
| maximal quotient of | #72293146, #72293163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36406);